Question

the following function is given. f(x)=x^3 - 5x^2 - 4x + 20 a. list all rational zeros that are possible according to the rational zero theorem. ±1,±2,±4,±5,±10,±20 (use a comma to separate answers as needed.) b. use synthetic division to test several possible rational zeros in order to identify one actual zero. one rational zero of the given function is 5 (simplify your answer.) c. use the zero from part (b) to find all the zeros of the polynomial function. the zeros of the function f(x)=x^3 - 5x^2 - 4x + 20 are (simplify your answer. type an integer or a fraction. use a comma to separate answers as needed.)

Explanation:

Step1: Recall synthetic division result

Since 5 is a zero, we divide $f(x)=x^{3}-5x^{2}-4x + 20$ by $(x - 5)$ using synthetic - division. The quotient is $x^{2}-4$.

Step2: Factor the quotient

We factor $x^{2}-4$ as a difference of squares. Using the formula $a^{2}-b^{2}=(a + b)(a - b)$, where $a=x$ and $b = 2$, we get $x^{2}-4=(x + 2)(x - 2)$.

Step3: Find all zeros

Set $(x + 2)(x - 2)(x - 5)=0$. Then, by the zero - product property, $x+2=0$ gives $x=-2$, $x - 2=0$ gives $x = 2$, and $x - 5=0$ gives $x = 5$.

Answer:

$5,2,-2$