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> | ||
+ | <cmath>p(a) = r(a)</cmath> | ||
− | == | + | ==Extension== |
− | + | An extension of the Remainder Theorem could be used to find the remainder of a polynomial when it is divided by a non-linear polynomial. Note that if <math>p(x)</math> is a polynomial, <math>q(x)</math> is the quotient, <math>d(x)</math> is a divisor, and <math>r(x)</math> is the remainder, the polynomial can be written as | |
+ | <cmath>p(x) = q(x)d(x) + r(x)</cmath> | ||
+ | Note that the degree of <math>r(x)</math> is less than the degree of <math>d(x)</math>. Let <math>a_n</math> be a root of <math>d(x)</math>, where <math>n</math> is an integer and <math>1 \le n \le \text{deg } d</math>. That means for all <math>a_n</math>, | ||
+ | <cmath>p(a_n) = r(a_n)</cmath> | ||
+ | Thus, the points <math>(a_n,p(a_n))</math> are on the graph of the remainder. If all the roots of <math>d(x)</math> are unique, then a [[system of equations]] can be made to find the remainder <math>r(x)</math>. | ||
+ | ==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]] | ||
+ | * [[1950 AHSME Problems/Problem 20]] | ||
− | + | ===Intermediate=== | |
+ | * [[1969 AHSME Problems/Problem 34]] | ||
+ | |||
+ | [[Category:Polynomials]] | ||
+ | [[Category:Algebra]] |
Revision as of 13:12, 4 December 2020
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
Extension
An extension of the Remainder Theorem could be used to find the remainder of a polynomial when it is divided by a non-linear polynomial. Note that if is a polynomial, is the quotient, is a divisor, and is the remainder, the polynomial can be written as Note that the degree of is less than the degree of . Let be a root of , where is an integer and . That means for all , Thus, the points are on the graph of the remainder. If all the roots of are unique, then a system of equations can be made to find the remainder .
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 . .