Question

20 multiple choice 1 point
which is a factor of $5x^2 - 2x - 7$?
$5x + 1$
$x + 7$
$5x - 1$
$5x + 7$
$5x - 7$
$x - 7$
$x - 1$
clear my selection
21 fill in the blank 1 point
the total number of solutions of a polynomial is determined by the of the polynomial.
type your answer...
22 multiple answer 1 point
list the possible zeros. choose all that apply.
$f(x) = 2x^3 + 7x^2 - 28x + 30$
$\pm \frac{1}{10}$
$\pm \frac{1}{4}$
$\pm \frac{2}{3}$
$\pm \frac{1}{2}$
$\pm 5$

Explanation:

Step1: Factor quadratic polynomial

We factor $5x^2 - 2x -7$. Find two numbers: $5$ and $-7$ (product $5\times(-7)=-35$, sum $5+(-7)=-2$). Rewrite:
$5x^2 +5x -7x -7 = 5x(x+1)-7(x+1)=(5x-7)(x+1)$? No, correction: $5x^2-2x-7=(5x+7)(x-1)$. So factor is $5x+7$.

Step2: Identify polynomial solution rule

By Fundamental Theorem of Algebra, total solutions equal polynomial's degree.

Step3: Apply Rational Root Theorem

For $f(x)=2x^3+7x^2-28x+30$, possible zeros are $\pm\frac{\text{factors of constant term }30}{\text{factors of leading coefficient }2}$. Factors of 30: $\pm1,\pm2,\pm3,\pm5,\pm6,\pm10,\pm15,\pm30$; factors of 2: $\pm1,\pm2$. Simplify to get $\pm\frac{1}{2},\pm1,\pm\frac{3}{2},\pm2,\pm3,\pm5,\pm\frac{15}{2},\pm6,\pm10,\pm15,\pm30$. Matching options: $\pm\frac{1}{2}$, $\pm5$.

Answer:

1. 5x+7
2. degree
3. $\pm\frac{1}{2}$, $\pm5$