Question

1. list all of the potential rational roots of $2x^5 - 7x^4 - 5x^3 + 18x^2 - 1$.

-1
2
$\frac{1}{2}$
1
$-\frac{1}{2}$
-2

1. list all of the potential rational roots of $x^5 - 7x^4 - 5x^3 + 18x^2 - 1$.

-7
$\frac{1}{7}$
-1
7
1

1. evaluate the polynomial $8x^3 + 15x^2 - 7x - 5$ at $x = 1$ using synthetic division.

1
-11
11
8

1. evaluate the polynomial $2x^5 - 7x^4 - 5x^3 + 18x^2 - 1$ at $x = -1$.

$\frac{1}{2}$
$-\frac{1}{2}$
-13
13

Explanation:

Step1: Apply Rational Root Theorem (Q5)

Potential roots: $\pm\frac{\text{factors of constant term}}{\text{factors of leading coefficient}}$
Constant term: $-1$, factors: $\pm1$; Leading coefficient: $2$, factors: $\pm1, \pm2$
Potential roots: $\pm1, \pm\frac{1}{2}$

Step2: Apply Rational Root Theorem (Q6)

Potential roots: $\pm\frac{\text{factors of constant term}}{\text{factors of leading coefficient}}$
Constant term: $-1$, factors: $\pm1$; Leading coefficient: $1$, factors: $\pm1$
Potential roots: $\pm1$

Step3: Synthetic division for Q7

Set up synthetic division for $8x^3 + 15x^2 -7x -5$ at $x=1$:

$$\begin{array}{r|rrrr} 1 & 8 & 15 & -7 & -5 \\ & & 8 & 23 & 16 \\ \hline & 8 & 23 & 16 & 11 \\ \end{array}$$

The remainder is the polynomial value.

Step4: Substitute $x=-1$ for Q8

Substitute $x=-1$ into $2x^5 -7x^4 -5x^3 +18x^2 -1$:
$2(-1)^5 -7(-1)^4 -5(-1)^3 +18(-1)^2 -1 = 2(-1) -7(1) -5(-1) +18(1) -1$
$=-2 -7 +5 +18 -1 = 13$

Answer:

1. $\boldsymbol{-1}$, $\boldsymbol{1}$, $\boldsymbol{-\frac{1}{2}}$, $\boldsymbol{\frac{1}{2}}$
2. $\boldsymbol{-1}$, $\boldsymbol{1}$
3. $\boldsymbol{11}$
4. $\boldsymbol{13}$