Answer to Question #158695 in Algebra for moon

Question

Use the Remainder Theorem to determine the remainder when P(x) is divided by D(x). Determine if D(x) is a factor of P(x).

1. P(x) = x³ - x² + x - 1; D(x) = x - 1
2. P(x) = 20x³ - 17x² + 6x - 26; D(x) = x + 1
3. P(x) = 2x³ - 5x + x² + 2; D(x) = x + 2
4. P(x) = 12x³ - 8x² + 4x - 3; D(x) = 2x - 1

Answer 1

By The Remainder Theorem:

When we divide a polynomial $P(x)$  by $(x-c)$ the remainder is $P(c)$

1. $P(x)=x^3-x^2+x-1;D(x)=x-1$

$$P(1)=(1)^3-(1)^2+1-1=0$$

$D(x)$ is a factor of $P(x).$

2. $P(x)=20x^3-17x^2+6x-26;D(x)=x+1$

$$P(-1)=20(-1)^3-17(-1)^2+6(-1)-26=-69$$

Remainder is $-69.$

3. $P(x)=2x^3+x^2-5x+2;D(x)=x+2$

$$P(-2)=2(-2)^3+(-2)^2-5(-2)+2=0$$

$D(x)$ is a factor of $P(x).$

4. $P(x)=12x^3-8x^2+4x-3;D(x)=2x-1$

$P_1(x)=6x^3-4x^2+2x-1.5$ divided by $D_1=(x-0.5)$

$$P_1(0.5)=6(0.5)^3-4(0.5)^2+2(0.5)-1.5=-0.75$$

Remainder is $2(-0.75)=-1.5.$