Question

Use the Factor Theorem to examine the polynomial p(x)=6x4+x3−45x2+26x+24. Which binomial is a factor of p(x)?

Answer 1

Answer:

(D) 3x−4

Step-by-step explanation:

Factor Theorem

Given a polynomial P(x) and a linear function x-a, If P(a)=0, then the linear function x-a is a factor of P(a).

In Option A:

$L inear Function =2x+3\\Set 2x+3=0 \\x=-\frac{3}{2} \\p(x)=6x^4+x^3-45x^2+26x+24\\p(-\frac{3}{2})=6(-\frac{3}{2})^4+(-\frac{3}{2})^3-45(-\frac{3}{2})^2+26(-\frac{3}{2})+24\\\\p(-\frac{3}{2})=-89.25$

In Option B

$L inear Function =3x-2\\Set 3x-2=0 \\x=\frac{2}{3} \\p(x)=6x^4+x^3-45x^2+26x+24\\p(\frac{2}{3})=6(\frac{2}{3})^4+(\frac{2}{3})^3-45(\frac{2}{3})^2+26(\frac{2}{3})+24\\\\p(\frac{2}{3})=22.8$

In Option C

$L inear Function =2x-1\\Set 2x-1=0 \\x=\frac{1}{2} \\p(x)=6x^4+x^3-45x^2+26x+24\\p(\frac{1}{2} )=6(\frac{1}{2} )^4+(\frac{1}{2} )^3-45(\frac{1}{2} )^2+26(\frac{1}{2} )+24\\\\p(\frac{1}{2} )=26.25$

In Option D

$L inear Function =3x-4\\Set 3x-4=0 \\x=\frac{4}{3} \\p(x)=6x^4+x^3-45x^2+26x+24\\p(\frac{4}{3} )=6(\frac{4}{3})^4+(\frac{4}{3} )^3-45(\frac{4}{3} )^2+26(\frac{4}{3})+24\\\\p(\frac{4}{3} )=0$

We can see that only Option D: 3x−4 gives a result of 0. Therefore, by the factor theorem, it is a factor of the polynomial.