Answer to Question #267307 in Calculus for Rama

Question

Is the term (x+2) a factor of the polynomial shown below? f(x)=x^4+5x^3+10x^2+20x+24

Answer 1

$$f(x)=x^4+5x^3+10x^2+20x+24$$

$$f(-2)=(-2)^4+5(-2)^3+10(-2)^2+20(-2)+24$$

$$=16-40+40-40+24=0$$

Then the term $(x+2)$ is a factor of the polynomial.

$$f(x)=x^4+5x^3+10x^2+20x+24$$

$$f(x)=x^4+2x^3+3x^3+6x^2+4x^2+8x+12x+24$$

$$=x^3(x+2)+3x^2(x+2)+4x(x+2)+12(x+2)$$

$$=(x+2)(x^3+3x^2+4x+12)$$

Therefore the term $(x+2)$ is a factor of the polynomial.