Question

H(x)=x-1/2 is a factor of f(x)=2x^4+x^3+x-3/4

Answer 1

Answer:

$\large{\text{Yes.}\ f(x)=x-\dfrac{1}{2} \text{is a factor of}\ f(x)=2x^4+x^3+x-\dfrac{3}{4}.}$

Step-by-step explanation:

$\text{If}\ (x-a)\ \text{is a factor of}\ p(x),\ \text{then}\ p(a)=0.$

$\text{Thereofre if}\ h(x)=x-\dfrac{1}{2}\ \text{is a factor of}\ f(x)=2x^4+x^3+x-\dfrac{3}{4},\ \text{then}\ f\left(\dfrac{1}{2}\right)=0.$

$\text{Substitute:}\\\\f\left(\dfrac{1}{2}\right)=2\left(\dfrac{1}{2}\right)^4+\left(\dfrac{1}{2}\right)^3+\dfrac{1}{2}-\dfrac{3}{4}=2\left(\dfrac{1}{16}\right)+\dfrac{1}{8}+\dfrac{1}{2}-\dfrac{3}{4}\\\\=\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1\cdot2}{2\cdot2}-\dfrac{3}{4}=\dfrac{1+1}{8}+\dfrac{2}{4}-\dfrac{3}{4}=\dfrac{2}{8}+\dfrac{2}{4}-\dfrac{3}{4}\\\\=\dfrac{1}{4}+\dfrac{2}{4}-\dfrac{3}{4}=\dfrac{1+2-3}{4}=\dfrac{0}{4}=0$