Remainder and Factor Theorem

Remainder Theorem and Factor Theorem

If a polynomial $F (x)$ divided by $p (x)$ , the quotient is $H (x)$ and the remainder is $S (x)$ , then the polynomial can be written as:

$F (x) = p (x) . H (x) + S (x)$

Example 01

Find the quotient and the remainder of $2 x^{3} + 5 x^{2} - 6 x + 3$ divided by $(x + 2)$

$\begin{array}{r} 2 x^{2} + x - 8 \\ x + 2 2 x^{3} + 5 x^{2} - 6 x + 3 \\ \underset{―}{2 x^{3} + 4 x^{2}} \\ x^{2} - 6 x + 3 \\ \underset{―}{x^{2} + 2 x} \\ - 8 x + 3 \\ \underset{―}{- 8 x - 16} \\ 19 \end{array}$

Quotient: $2 x^{2} + x - 8$

Remainder: $19$

$2 x^{3} + 5 x^{2} - 6 x + 3 = (x + 2) . (2 x^{2} + x - 8) + 19$

Remainder Theorem

If a polynomial $F (x)$ divided by $(x - a)$ , the remainder is $F (a)$ .

Factor Theorem

If $(x - a)$ is a factor of a polynomial $F (x)$ , then $F (a) = 0$ .

Example 02

Find the remainder of $F (x) = x^{3} + 4 x^{2} - 6 x + 1$ divided by $(x - 2)$

$F (x) = x^{3} + 4 x^{2} - 6 x + 1$

$F (2) = 2^{3} + 4 . 2^{2} - 6 . 2 + 1$

$F (2) = 13$

We can write the polynomial as

$x^{3} + 4 x^{2} - 6 x + 1 = (x - 2) . H (x) + 13$

Example 03

Prove that $(x + 3)$ is a factor of $F (x) = x^{3} - x^{2} - 10 x + 6$

$F (- 3) = (- 3)^{3} - (- 3)^{2} - 10 (- 3) + 6 = 0$

Since $F (- 3) = 0$ , then $(x + 3)$ is a factor of $F (x)$ .

Exercise

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Remainder Theorem and Factor Theorem

Exercise

Algebra