Remainder theorem - Polynomial Functions

Remainder theorem

You may want to refresh your memory on polynomial long division and synthetic division to better understand this lesson. The remainder theorem simply states that if a polynomial f(x) is divided by a linear expression x-r, the value of f(r) is equal to the remainder.

Basic concepts:
Polynomial long division
Polynomial synthetic division

Related concepts:
Integration of rational functions by partial fractions

Lessons

Notes:

$\cdot$ When a polynomial, $P(x)$ , is divided by $(x-a)$ : Remainder $=P(a)$
$\cdot$ When a polynomial, $P(x)$ , is divided by $(ax-b)$ : Remainder $=P(\frac{b}{a})$

2.
Finding the Remainder Using Synthetic Division and the Remainder Theorem
Find the remainder when $\left( {4{x^3} - 7x + 10} \right)$ is divided by $\left( {2x - 5} \right)$

Remainder theorem

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Notes:

⋅\cdot⋅ When a polynomial, P(x)P(x)P(x), is divided by (x−a)(x-a)(x−a): Remainder =P(a)=P(a)=P(a) ⋅\cdot⋅ When a polynomial, P(x)P(x)P(x), is divided by (ax−b)(ax-b)(ax−b): Remainder =P(ba)=P(\frac{b}{a})=P(ab​)

1.

Understanding the remainder Theorem Prove the Remainder Theorem

2.

Finding the Remainder Using Synthetic Division and the Remainder Theorem Find the remainder when (4x3−7x+10)\left( {4{x^3} - 7x + 10} \right)(4x3−7x+10) is divided by (2x−5)\left( {2x - 5} \right)(2x−5)

a)

Using synthetic division

b)

Using the remainder theorem

3.

When (8x3+ax2+bx−1)\left( {8{x^3} + a{x^2} + bx - 1} \right)(8x3+ax2+bx−1) is divided by: i) (2x−5)\left( {2x - 5} \right)(2x−5), the remainder is 545454 ii) (x+1)\left( {x + 1} \right)(x+1), the remainder is −30 - 30−30 Find the values of aaa and bbb.

Remainder theorem

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Polynomial long division

Algebra
Polynomial synthetic division

or

$\cdot$ When a polynomial, $P(x)$ , is divided by $(x-a)$ : Remainder $=P(a)$
$\cdot$ When a polynomial, $P(x)$ , is divided by $(ax-b)$ : Remainder $=P(\frac{b}{a})$

Understanding the remainder Theorem
Prove the Remainder Theorem

Finding the Remainder Using Synthetic Division and the Remainder Theorem
Find the remainder when $\left( {4{x^3} - 7x + 10} \right)$ is divided by $\left( {2x - 5} \right)$

When $\left( {8{x^3} + a{x^2} + bx - 1} \right)$ is divided by:
i) $\left( {2x - 5} \right)$ , the remainder is $54$
ii) $\left( {x + 1} \right)$ , the remainder is $- 30$
Find the values of $a$ and $b$ .