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SYNTHETIC DIVISION
Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor -- and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials.CRAMER'S RULE
Evaluating each determinant (using the method explained here), we get:. Cramer's Rule says that x = D x ÷ D, y = D y ÷ D, and z = D z ÷ D.That is: x = 3 / 3 = 1, y = –6 / 3 = –2, and z = 9 / 3 = 3. That's all there is to Cramer's Rule. To find whichever variable youwant (call it
RIGHT-TRIANGLE WORD PROBLEMS 165/143 = tan (θ) tan–1 (165/143) = θ = 49.08561678 But this is not the "bearing", since the bearing is the angle with respect to "due north". I need to add in the original forty-degree angle to get my answer: The plane is about 218 miles away, at a bearing of about 89°. Another major class of right-triangle word problems you will ADDING AND SUBTRACTING MATRICES Demonstrates how to add and subtract matrices, explains why the addition or subtraction sometimes can't be done, and gives an example of how matrix addition is used in homework problems. ADDING & SUBTRACTING RADICALS (SQUARE ROOTS) Explains how to combine 'like' radical terms when adding and subtracting expressions containing square roots. Provides workedexamples.
EXPONENTS: BASIC RULES Purplemath. Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 5 3.The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being 5 in this THE CHANGE-OF-BASE FORMULA Purplemath. There is one other log "rule", but it's more of a formula than a rule. You may have noticed that your calculator only has keys for figuring the values for the common (that is, the base-10) log and the natural (that is, the base-e) log.There are no keys for any otherbases.
PARTIAL-FRACTION DECOMPOSITION: GENERAL TECHNIQUES Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions. To decompose a fraction, you first factor the denominator. Let's work backwards from the example above. The denominator is x2 + x, which factors as x ( x +1).
PURPLEMATH | HOMEIN STUDY ORDERTRIGONOMETRIC IDENTITIESPRACTICAL ALGEBRA LESSONSQUADRATIC FORMULARADICALS Purplemath's " Homework Guidelines for Mathematics " will give you a leg up, explaining in clear terms what your math teacher is looking for. The Guidelines link to examples of common errors, and demonstrate techniques that your instructors will love! In addition, students who get in the habit of explaining themselves clearly in their homework TRIGONOMETRIC IDENTITIES Purplemath. In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a 2 + b 2 = c 2" for right triangles.There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.SYNTHETIC DIVISION
Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor -- and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials.CRAMER'S RULE
Evaluating each determinant (using the method explained here), we get:. Cramer's Rule says that x = D x ÷ D, y = D y ÷ D, and z = D z ÷ D.That is: x = 3 / 3 = 1, y = –6 / 3 = –2, and z = 9 / 3 = 3. That's all there is to Cramer's Rule. To find whichever variable youwant (call it
RIGHT-TRIANGLE WORD PROBLEMS 165/143 = tan (θ) tan–1 (165/143) = θ = 49.08561678 But this is not the "bearing", since the bearing is the angle with respect to "due north". I need to add in the original forty-degree angle to get my answer: The plane is about 218 miles away, at a bearing of about 89°. Another major class of right-triangle word problems you will ADDING AND SUBTRACTING MATRICES Demonstrates how to add and subtract matrices, explains why the addition or subtraction sometimes can't be done, and gives an example of how matrix addition is used in homework problems. ADDING & SUBTRACTING RADICALS (SQUARE ROOTS) Explains how to combine 'like' radical terms when adding and subtracting expressions containing square roots. Provides workedexamples.
EXPONENTS: BASIC RULES Purplemath. Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 5 3.The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being 5 in this THE CHANGE-OF-BASE FORMULA Purplemath. There is one other log "rule", but it's more of a formula than a rule. You may have noticed that your calculator only has keys for figuring the values for the common (that is, the base-10) log and the natural (that is, the base-e) log.There are no keys for any otherbases.
PARTIAL-FRACTION DECOMPOSITION: GENERAL TECHNIQUES Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions. To decompose a fraction, you first factor the denominator. Let's work backwards from the example above. The denominator is x2 + x, which factors as x ( x +1).
BASIC LOG RULES & EXPANDING LOG EXPRESSIONS Warning: Just as when you're dealing with exponents, the above rules work only if the bases are the same. For instance, the expression "log d (m) + log b (n)" cannot be simplified, because the bases (the "d" and the "b") are not the same, just as x 2 × y 3 cannot be simplified because the bases (the xSET NOTATION
Purplemath. You never know when set notation is going to pop up. Usually, you'll see it when you learn about solving inequalities, because for some reason saying "x < 3" isn't good enough, so instead they'll want you to phrase the answer as "the solution set is { x | x is a real number and x < 3 }".How this adds anything to the student's understanding, I don't know. MEAN, MEDIAN, MODE, AND RANGE There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5 th number: 13, 13, 13, 13, 14, 14, 16, 18, 21. 13, 13, 13, 13, 14, 14, 16, 18, 21. So the median is 14. The mode is the number that is repeated more often than any other, so 13 is the mode. The largest value in the list is 21, and the smallest is 13, so ADDING AND SUBTRACTING MATRICES Demonstrates how to add and subtract matrices, explains why the addition or subtraction sometimes can't be done, and gives an example of how matrix addition is used in homework problems. ANGULAR AND LINEAR VELOCITY, AND RPM Purplemath. For some reason, it seems fairly common for textbooks to turn to issues of angular velocity, linear velocity, and revolutions per minute (rpm) shortly after explaining circle sectors, their areas, and their arc lengths.An arc's length is the distance partway around a circle; and the linear distance covered by, say, a bicycle is related to the radius of the bike's tires. GRAPHING TRIGONOMETRIC FUNCTIONS Purplemath. You've already learned the basic trig graphs.But just as you could make the basic quadratic, y = x 2, more complicated, such as y = –(x + 5) 2 – 3, so also trig graphs can be made more complicated.We can transform and translate trig functions, just like you transformed and translated other functions in algebra.. Let's start with the basic sine function, f (t) = sin(t). "WORK" WORD PROBLEMS Purplemath. Another "typical" work problem is the "one guy did part of the job" or "the number of workers changed at some point during the job" type. We'll still need to do the computations for how much each guy does per unit time (usually hours or days), but we may need to use the fact that "a completed task" is represented by " 1 ". DESCARTES' RULE OF SIGNS Use Descartes' Rule of Signs to find the number of real roots of:f (x) = x5 + 4x4 – 3x2 + x – 6. First, I look at the positive-root case, which is looking at f (x): f ( x) = +x5 + 4 x4 – 3 x2 + x – 6. The signs flip three times, so there are three positive roots, or one positive root. Either way, I THE DISTANCE FORMULA Explains the Distance Formula, how the Distance Formula is derived from the Pythagorean Theorem, and how to use the Formula. SPECIAL ANGLE VALUES: 30-60-90 AND 45-45-90 TRIANGLES Explains a simple pictorial way to remember basic reference angle values. Provides other memory aids for the values of trigonometric ratios for these "special" angle values, based on PURPLEMATH | HOMEIN STUDY ORDERTRIGONOMETRIC IDENTITIESPRACTICAL ALGEBRA LESSONSQUADRATIC FORMULARADICALS Purplemath's " Homework Guidelines for Mathematics " will give you a leg up, explaining in clear terms what your math teacher is looking for. The Guidelines link to examples of common errors, and demonstrate techniques that your instructors will love! In addition, students who get in the habit of explaining themselves clearly in their homework TRIGONOMETRIC IDENTITIES Purplemath. In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a 2 + b 2 = c 2" for right triangles.There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.CRAMER'S RULE
Evaluating each determinant (using the method explained here), we get:. Cramer's Rule says that x = D x ÷ D, y = D y ÷ D, and z = D z ÷ D.That is: x = 3 / 3 = 1, y = –6 / 3 = –2, and z = 9 / 3 = 3. That's all there is to Cramer's Rule. To find whichever variable youwant (call it
ADDING AND SUBTRACTING MATRICES Demonstrates how to add and subtract matrices, explains why the addition or subtraction sometimes can't be done, and gives an example of how matrix addition is used in homework problems. SOLVING EXPONENTIAL EQUATIONS WITH LOGARITHMS Purplemath. Most exponential equations do not solve neatly; there will be no way to convert the bases to being the same, such as the conversion of 4 and 8 into powers of 2. In solving these more-complicated equations, you will have to use logarithms. Taking logarithms will allow us to take advantage of the log rule that saysthat powers inside
THE DISTANCE FORMULA Explains the Distance Formula, how the Distance Formula is derived from the Pythagorean Theorem, and how to use the Formula. CONDENSING LOG EXPRESSIONS Purplemath. The logs rules work "backwards", so you can condense ("compress"?) strings of log expressions into one log with a complicated argument. When they tell you to "simplify" a log expression, this usually means they will have given you lots of log terms, each containing a simple argument, and they want you to combineeverything into one
THE CHANGE-OF-BASE FORMULA Purplemath. There is one other log "rule", but it's more of a formula than a rule. You may have noticed that your calculator only has keys for figuring the values for the common (that is, the base-10) log and the natural (that is, the base-e) log.There are no keys for any otherbases.
ADDING & SUBTRACTING RADICALS (SQUARE ROOTS) Explains how to combine 'like' radical terms when adding and subtracting expressions containing square roots. Provides workedexamples.
SPECIAL ANGLE VALUES: 30-60-90 AND 45-45-90 TRIANGLESSEE MORE ONPURPLEMATH.COM
PURPLEMATH | HOMEIN STUDY ORDERTRIGONOMETRIC IDENTITIESPRACTICAL ALGEBRA LESSONSQUADRATIC FORMULARADICALS Purplemath's " Homework Guidelines for Mathematics " will give you a leg up, explaining in clear terms what your math teacher is looking for. The Guidelines link to examples of common errors, and demonstrate techniques that your instructors will love! In addition, students who get in the habit of explaining themselves clearly in their homework TRIGONOMETRIC IDENTITIES Purplemath. In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a 2 + b 2 = c 2" for right triangles.There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.CRAMER'S RULE
Evaluating each determinant (using the method explained here), we get:. Cramer's Rule says that x = D x ÷ D, y = D y ÷ D, and z = D z ÷ D.That is: x = 3 / 3 = 1, y = –6 / 3 = –2, and z = 9 / 3 = 3. That's all there is to Cramer's Rule. To find whichever variable youwant (call it
ADDING AND SUBTRACTING MATRICES Demonstrates how to add and subtract matrices, explains why the addition or subtraction sometimes can't be done, and gives an example of how matrix addition is used in homework problems. SOLVING EXPONENTIAL EQUATIONS WITH LOGARITHMS Purplemath. Most exponential equations do not solve neatly; there will be no way to convert the bases to being the same, such as the conversion of 4 and 8 into powers of 2. In solving these more-complicated equations, you will have to use logarithms. Taking logarithms will allow us to take advantage of the log rule that saysthat powers inside
THE DISTANCE FORMULA Explains the Distance Formula, how the Distance Formula is derived from the Pythagorean Theorem, and how to use the Formula. CONDENSING LOG EXPRESSIONS Purplemath. The logs rules work "backwards", so you can condense ("compress"?) strings of log expressions into one log with a complicated argument. When they tell you to "simplify" a log expression, this usually means they will have given you lots of log terms, each containing a simple argument, and they want you to combineeverything into one
THE CHANGE-OF-BASE FORMULA Purplemath. There is one other log "rule", but it's more of a formula than a rule. You may have noticed that your calculator only has keys for figuring the values for the common (that is, the base-10) log and the natural (that is, the base-e) log.There are no keys for any otherbases.
ADDING & SUBTRACTING RADICALS (SQUARE ROOTS) Explains how to combine 'like' radical terms when adding and subtracting expressions containing square roots. Provides workedexamples.
SPECIAL ANGLE VALUES: 30-60-90 AND 45-45-90 TRIANGLESSEE MORE ONPURPLEMATH.COM
CONVERTING UNITS: EXAMPLES The conversion ratios are 1 acre = 43,560 ft2, 1ft3 = 7.481 gallons, and five gallons = 1 water bottle. First I have to figure out the volume in one acre-foot. An acre-foot is the amount that it would take to cover one acre of land to a depth of one foot. BASIC LOG RULES & EXPANDING LOG EXPRESSIONS Warning: Just as when you're dealing with exponents, the above rules work only if the bases are the same. For instance, the expression "log d (m) + log b (n)" cannot be simplified, because the bases (the "d" and the "b") are not the same, just as x 2 × y 3 cannot be simplified because the bases (the x DESCARTES' RULE OF SIGNS Use Descartes' Rule of Signs to find the number of real roots of:f (x) = x5 + 4x4 – 3x2 + x – 6. First, I look at the positive-root case, which is looking at f (x): f ( x) = +x5 + 4 x4 – 3 x2 + x – 6. The signs flip three times, so there are three positive roots, or one positive root. Either way, I SOLVING HARDER TRIG EQUATIONS The first solution is 45° more than a multiple of 180°, so (180n)° + 45° should do. The second solution is 30° more than a multiple of 180° and (because of the "plus / minus") also 30° less than that same multiple, so (180n)° ± 30° will cover this part. x = (180n)° 30°, (180n)° + 45° for all integers n. GRAPHING TRIGONOMETRIC FUNCTIONS Purplemath. You've already learned the basic trig graphs.But just as you could make the basic quadratic, y = x 2, more complicated, such as y = –(x + 5) 2 – 3, so also trig graphs can be made more complicated.We can transform and translate trig functions, just like you transformed and translated other functions in algebra.. Let's start with the basic sine function, f (t) = sin(t). SOLVING SIMPLE (TO MEDIUM-HARD) TRIG EQUATIONS Purplemath. Solving trig equations use both the reference angles and trigonometric identities that you've memorized, together with a lot of the algebra you've learned. Be prepared to need to think in order to solve these equations.. In what follows, it is assumed that you have a good grasp of the trig-ratio values in the first quadrant, how the unit circle works, the relationship between QUADRATIC "MAX/MIN" WORD PROBLEMS When you get to calculus, you will see some of these max/min exercises again. At that point, they'll want you to differentiate to find the maximums and minimums; at this point, you'll find the vertex, since the vertex will be the maximum or minimum of the related graphed parabola.But they're the same exercise and you'll get the same answers then as you will now. FACTORING QUADRATICS: THE HARD CASE Purplemath. A "hard" quadratic is one whose leading coefficient (that is, whose numerical value on the x 2 term) is something other than a nice, well-behaved 1.To factor a "hard" quadratic, we have to handle all three coefficients, not just the two we handled in the "easy" case, because the leading coefficient adds to the mix, and makes things much messier. FUNCTION NOTATION: DEFINITIONS & EVALUATING AT A NUMBER In the same way, in textbooks and when writing things out, we use different function names like f (x), g(x), h(x), s(t), etc, to keep track of, and work with, more than one formula in any single context.With function notation, we can now use more than one function at a time without confusing ourselves or mixing up the formulas, leaving ourselves wondering "Okay, which ' y ' is this one?" LINEAR PROGRAMMING: HOW TO SET UP WORD PROBLEMS Given the inequalities, linear-programming exercise are pretty straightforward, if sometimes a bit long. The hard part is usually the word problems, where you have to figure out what the inequalities are. So I'll show how to set up some typical linear-programming wordproblems.
PURPLEMATH | HOMEIN STUDY ORDERTRIGONOMETRIC IDENTITIESPRACTICAL ALGEBRA LESSONSQUADRATIC FORMULARADICALS Purplemath's " Homework Guidelines for Mathematics " will give you a leg up, explaining in clear terms what your math teacher is looking for. The Guidelines link to examples of common errors, and demonstrate techniques that your instructors will love! In addition, students who get in the habit of explaining themselves clearly in their homework TRIGONOMETRIC IDENTITIES Purplemath. In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a 2 + b 2 = c 2" for right triangles.There are loads of trigonometric identities, but the following are the ones you're most likely to see and use. MEAN, MEDIAN, MODE, AND RANGE There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5 th number: 13, 13, 13, 13, 14, 14, 16, 18, 21. 13, 13, 13, 13, 14, 14, 16, 18, 21. So the median is 14. The mode is the number that is repeated more often than any other, so 13 is the mode. The largest value in the list is 21, and the smallest is 13, so POLYNOMIALS: DEFINITIONS & EVALUATION EVEN AND ODD FUNCTIONS Purplemath. You may be asked to "determine algebraically" whether a function is even or odd. To do this, you take the function and plug –x in for x, and then simplify.If you end up with the exact same function that you started with (that is, if f (–x) = f (x), so all of the signs are the same), then the function is even.If you end up with the exact opposite of what you started with (that "WORK" WORD PROBLEMS Purplemath. Another "typical" work problem is the "one guy did part of the job" or "the number of workers changed at some point during the job" type. We'll still need to do the computations for how much each guy does per unit time (usually hours or days), but we may need to use the fact that "a completed task" is represented by " 1 ". DESCARTES' RULE OF SIGNS Use Descartes' Rule of Signs to find the number of real roots of:f (x) = x5 + 4x4 – 3x2 + x – 6. First, I look at the positive-root case, which is looking at f (x): f ( x) = +x5 + 4 x4 – 3 x2 + x – 6. The signs flip three times, so there are three positive roots, or one positive root. Either way, ICRAMER'S RULE
Evaluating each determinant (using the method explained here), we get:. Cramer's Rule says that x = D x ÷ D, y = D y ÷ D, and z = D z ÷ D.That is: x = 3 / 3 = 1, y = –6 / 3 = –2, and z = 9 / 3 = 3. That's all there is to Cramer's Rule. To find whichever variable youwant (call it
RADICALS: INTRODUCTION & SIMPLIFICATION Note that the value of the simplified radical is positive.While either of +2 and –2 might have been squared to get 4, "the square root of four" is defined to be only the positive option, +2.That is, the definition of the square root says that the square root will spit out only the positive root.. On a side note, let me emphasize that "evaluating" an expression (to find its one value) and ADDING & SUBTRACTING RADICALS (SQUARE ROOTS) Explains how to combine 'like' radical terms when adding and subtracting expressions containing square roots. Provides workedexamples.
PURPLEMATH | HOMEIN STUDY ORDERTRIGONOMETRIC IDENTITIESPRACTICAL ALGEBRA LESSONSQUADRATIC FORMULARADICALS Purplemath's " Homework Guidelines for Mathematics " will give you a leg up, explaining in clear terms what your math teacher is looking for. The Guidelines link to examples of common errors, and demonstrate techniques that your instructors will love! In addition, students who get in the habit of explaining themselves clearly in their homework TRIGONOMETRIC IDENTITIES Purplemath. In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a 2 + b 2 = c 2" for right triangles.There are loads of trigonometric identities, but the following are the ones you're most likely to see and use. MEAN, MEDIAN, MODE, AND RANGE There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5 th number: 13, 13, 13, 13, 14, 14, 16, 18, 21. 13, 13, 13, 13, 14, 14, 16, 18, 21. So the median is 14. The mode is the number that is repeated more often than any other, so 13 is the mode. The largest value in the list is 21, and the smallest is 13, so POLYNOMIALS: DEFINITIONS & EVALUATION EVEN AND ODD FUNCTIONS Purplemath. You may be asked to "determine algebraically" whether a function is even or odd. To do this, you take the function and plug –x in for x, and then simplify.If you end up with the exact same function that you started with (that is, if f (–x) = f (x), so all of the signs are the same), then the function is even.If you end up with the exact opposite of what you started with (that "WORK" WORD PROBLEMS Purplemath. Another "typical" work problem is the "one guy did part of the job" or "the number of workers changed at some point during the job" type. We'll still need to do the computations for how much each guy does per unit time (usually hours or days), but we may need to use the fact that "a completed task" is represented by " 1 ". DESCARTES' RULE OF SIGNS Use Descartes' Rule of Signs to find the number of real roots of:f (x) = x5 + 4x4 – 3x2 + x – 6. First, I look at the positive-root case, which is looking at f (x): f ( x) = +x5 + 4 x4 – 3 x2 + x – 6. The signs flip three times, so there are three positive roots, or one positive root. Either way, ICRAMER'S RULE
Evaluating each determinant (using the method explained here), we get:. Cramer's Rule says that x = D x ÷ D, y = D y ÷ D, and z = D z ÷ D.That is: x = 3 / 3 = 1, y = –6 / 3 = –2, and z = 9 / 3 = 3. That's all there is to Cramer's Rule. To find whichever variable youwant (call it
RADICALS: INTRODUCTION & SIMPLIFICATION Note that the value of the simplified radical is positive.While either of +2 and –2 might have been squared to get 4, "the square root of four" is defined to be only the positive option, +2.That is, the definition of the square root says that the square root will spit out only the positive root.. On a side note, let me emphasize that "evaluating" an expression (to find its one value) and ADDING & SUBTRACTING RADICALS (SQUARE ROOTS) Explains how to combine 'like' radical terms when adding and subtracting expressions containing square roots. Provides workedexamples.
PRACTICAL ALGEBRA LESSONS Pre-algebra and algebra lessons, from negative numbers through pre-calculus. Grouped by level of study. Lessons are practical in nature informal in tone, and contain many worked examples and warnings about problem areas and probable "trick" questions. RADICALS: INTRODUCTION & SIMPLIFICATION Note that the value of the simplified radical is positive.While either of +2 and –2 might have been squared to get 4, "the square root of four" is defined to be only the positive option, +2.That is, the definition of the square root says that the square root will spit out only the positive root.. On a side note, let me emphasize that "evaluating" an expression (to find its one value) and SOLVING EXPONENTIAL EQUATIONS WITH LOGARITHMS Purplemath. Most exponential equations do not solve neatly; there will be no way to convert the bases to being the same, such as the conversion of 4 and 8 into powers of 2. In solving these more-complicated equations, you will have to use logarithms. Taking logarithms will allow us to take advantage of the log rule that saysthat powers inside
VENN DIAGRAMS: SET NOTATION Purplemath. Venn diagrams can be used to express the logical (in the mathematical sense) relationships between various sets. The following examples should help you understand the notation, terminology, and concepts relating Venn diagrams and set notation.. Let's say that our universe contains the numbers 1, 2, 3, and 4, so U = {1, 2, 3, 4}.Let A be the set containing the numbers 1 and 2; that THE ORDER OF OPERATIONS: PEMDAS The "operations" are addition, subtraction, multiplication, division, exponentiation, and grouping; the "order" of these operations states which operations take precedence (are taken care of) before which other operations. A common technique for remembering the order of operations is the abbreviation (or, more properly, the "acronym")"PEMDAS
SOLVING LOG EQUATIONS WITH EXPONENTIALS log 2 ( x2 – 2 x) = 3. Now the equation is arranged in a useful way. At this point, I can use The Relationship to convert the log form of the equation to the corresponding exponential form, and then I can solve the result: log 2 ( x2 – 2 x) = 3. 2 3 = x2 – 2 x. 8 = x2 – 2 x. 0 = x2 – 2 x – 8. 0 = ( x – 4) ( x + 2) FUNCTION NOTATION: DEFINITIONS & EVALUATING AT A NUMBER In the same way, in textbooks and when writing things out, we use different function names like f (x), g(x), h(x), s(t), etc, to keep track of, and work with, more than one formula in any single context.With function notation, we can now use more than one function at a time without confusing ourselves or mixing up the formulas, leaving ourselves wondering "Okay, which ' y ' is this one?" HOW CAN 0.999... = 1? There are many different proofs of the fact that "0.9999" does indeed equal 1.So why does this question keep coming up? Students don't generally argue with "0.3333" being equal to 1 / 3, but then, one-third is a fraction.Maybe it's just that it "feels" "wrong" that something as nice and neat and well-behaved as the number "1" could also be written in such a messy form as "0.9999 SIGNIFICANT DIGITS: ADDITIONAL CONSIDERATIONS Find the product of 0.00435 and 4.6, simplifying to the appropriate number of digits. First I multiply: 0.00435 × 4.6 = 0.02001. Looking at the original numbers, I see that 4.6 has only two significant digits, so I will have to round 0.02001 to two significant digits. The MINORS AND COFACTORS: EXPANDING ALONG A ROW Minors and Cofactors: Introduction; Expanding Along a Row. The process for 3×3 matrices, while a bit messier, is still pretty straightforward: You add repeats of the first and second columns to the end of the determinant, multiply along all the diagonals, and add and subtract according to the rule: But for 4×4 's and biggerdeterminants, you
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Need help with math? Start browsing Purplemath's free resources below! PRACTIAL ALGEBRA LESSONS: Purplemath's algebra lessons are informal in their tone, and are written with the struggling student in mind. Don't worry about overly-professorial or confusing language! These math lessons emphasize the practicalities rather than the technicalities, demonstrating dependably helpful techniques, warning of likely "trick" test questions, and pointing out common student mistakes. Content Continues BelowMATHHELP.COM
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------------------------- ------------------------- HOMEWORK GUIDELINES: English teachers tell their students explicitly how to format their papers: what fonts, what page margins, what style guides, etc. Math teachers, on the other hand, frequently just complain amongst themselves in the faculty lounge about how messy their students' work is. Meanwhile, their students wonder why they've lost points on homework and tests. Neat homework can aid your comprehension and might make your teacher like you better. Purplemath's "Homework Guidelines for Mathematics " will give you a leg up, explaining in clear terms what your math teacher is looking for. The Guidelines link to examples of common errors, and demonstrate techniques that your instructors will love! In addition, students who get in the habit of explaining themselves clearly in their homework gain greater understanding of what they're doing, and therefore tend to do much better on their tests. Don't leave easy points on the table! Study these Guidelines , print out this "formatting " PDF, and improve your learning, retention, and test scores! Go to the Guidelines! ------------------------- STUDY SKILLS SELF-SURVEY: Many students, from time to time, wonder, "Do I have what it takes to succeed in math?" Much of one's success or failure in algebra can be laid at the feet of one's study habits. Do you have good math study habits? Take this survey andfind out.
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