Question
Consider the polynomial $f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e,$ where$a+b+c+d+e=0 .$ Show that thispolynomial is divisible by $x-1$
Step 1
Step 1: First, recall the factor theorem which states that if $f(a) = 0$ for some $a$, then $(x-a)$ is a factor of the polynomial $f(x)$. Show more…
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CHALLENGE. Consider the polynomial $f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e,$ where $a+b+c+d+e=0 .$ Show that this polynomial is divisible by $x-1$
Polynomial Functions
The Remainder and Factor Theorems
If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=x f\left(x^{3}\right)+x^{2} g\left(x^{6}\right)$ is divisible by $x^{2}+x+1$, then (A) $f(1)=g(1)$ (B) $f(1)=-g(1)$ (C) $h(1)=0$ (D) $h(-1)=0$
Use the Remainder Theorem to find the remainder when $f(x)$ is divided by $x-c .$ Then use the Factor Theorem to determine whether $x-c$ is a factor of $f(x)$ $$ f(x)=3 x^{4}+x^{3}-3 x+1 ; x+\frac{1}{3} $$
Polynomial and Rational Functions
The Real Zeros of a Polynomial Function
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