Question

1. use the constant term and leading coefficient of each expression to list all its potential roots.

$3x^5 - 7x^4 - 5x^3 + 18x^2 - 5$
factors of constant:
factors of leading coefficient:
potential roots:
a. $\pm1, \pm\frac{1}{3}, \pm5, \pm\frac{5}{3}$
b. $\pm1, \pm3$
c. $\pm\frac{3}{5}, \pm3$
d. $\pm1, \pm3, \pm5, \pm15$
e. $\pm1, \pm5$

1. choose the best answer.

the value of a polynomial at $x = 7$ is the ___ when the polynomial is divided by $x - 7$.
leading
remainder
constant
zero

1. choose the best answer.

the numerators of any rational roots of a polynomial will be the factors of the ___ term.
remainder
synthetic
constant
leading

Explanation:

Step1: Identify constant term

For polynomial $3x^5 - 7x^4 - 5x^3 + 18x^2 - 5$, constant term is $-5$. Its factors are $\pm1, \pm5$.

Step2: Identify leading coefficient

Leading coefficient is $3$. Its factors are $\pm1, \pm3$.

Step3: Apply Rational Root Theorem

Potential roots = $\frac{\text{factors of constant term}}{\text{factors of leading coefficient}} = \pm1, \pm\frac{1}{3}, \pm5, \pm\frac{5}{3}$.

Step4: Solve Q16 (Remainder Theorem)

By Remainder Theorem, $P(7)$ is remainder when $P(x)\div(x-7)$.

Step5: Solve Q17 (Rational Root Theorem)

Numerators of rational roots are factors of the constant term.

Answer:

Question 15:

factors of constant: $\pm1, \pm5$
factors of leading coefficient: $\pm1, \pm3$
potential roots: a. $\pm1, \pm\frac{1}{3}, \pm5, \pm\frac{5}{3}$

Question 16:

remainder

Question 17:

constant