Difference between revisions of "Remainder Theorem"
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− | =Theorem= | + | ==Theorem== |
The Remainder Theorem states that the remainder when the polynomial <math>P(x)</math> is divided by <math>x-a</math> (usually with synthetic division) is equal to the simplified value of <math>P(a)</math>. | The Remainder Theorem states that the remainder when the polynomial <math>P(x)</math> is divided by <math>x-a</math> (usually with synthetic division) is equal to the simplified value of <math>P(a)</math>. | ||
− | = | + | ==Proof== |
− | + | Let <math>\frac{p(x)}{x-a} = q(x) + \frac{r(x)}{x-a}</math>, where <math>p(x)</math> is the polynomial, <math>x-a</math> is the divisor, <math>q(x)</math> is the quotient, and <math>r(x)</math> is the remainder. This equation can be rewritten as | |
− | + | <cmath>p(x) = q(x) \cdot (x-a) + r(x)</cmath> | |
− | + | If <math>x = a</math>, then substituting for <math>x</math> results in | |
− | + | <cmath>p(a) = q(a) \cdot (a - a) + r(a)</cmath> | |
− | + | <cmath>p(a) = q(a) \cdot 0 + r(a)</cmath> | |
+ | <math></math>p(a) = r(a) | ||
+ | ==Examples== | ||
+ | ===Introductory=== | ||
+ | * What is the remainder when <math>x^2+2x+3</math> is divided by <math>x+1</math>? | ||
+ | ''Solution'': Using synthetic or long division we obtain the quotient <math>1+\frac{2}{x^2+2x+3}</math>. In this case the remainder is <math>2</math>. However, we could've figured that out by evaluating <math>P(-1)</math>. Remember, we want the divisor in the form of <math>x-a</math>. <math>x+1=x-(-1)</math> so <math>a=-1</math>. | ||
<math>P(-1) = (-1)^2+2(-1)+3 = 1-2+3 = \boxed{2}</math>. | <math>P(-1) = (-1)^2+2(-1)+3 = 1-2+3 = \boxed{2}</math>. | ||
+ | * [[1961 AHSME Problems/Problem 22]] | ||
{{stub}} | {{stub}} |
Revision as of 10:47, 20 May 2018
Contents
[hide]Theorem
The Remainder Theorem states that the remainder when the polynomial is divided by (usually with synthetic division) is equal to the simplified value of .
Proof
Let , where is the polynomial, is the divisor, is the quotient, and is the remainder. This equation can be rewritten as If , then substituting for results in $$ (Error compiling LaTeX. Unknown error_msg)p(a) = r(a)
Examples
Introductory
- What is the remainder when is divided by ?
Solution: Using synthetic or long division we obtain the quotient . In this case the remainder is . However, we could've figured that out by evaluating . Remember, we want the divisor in the form of . so . .
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