Question

answer the following questions.

1. choose the best answer.

the numerators of any rational roots of a polynomial will be factors of the ____.
constant term
degree of the polynomial
sum of the coefficients
leading coefficient

1. choose the best answer.

the denominators of any rational roots of a polynomial will be factors of the ____.
degree of the polynomial
sum of the coefficients
constant term
leading coefficient

1. choose the best answer.

the value of a polynomial at ( x = 1 ) is the remainder when the polynomial is divided by ____.
1
( x - 1 )
-1
( x + 1 )

1. which of these are not potential rational roots of ( 8x^3 + 15x^2 - 7x - 5 )?

( pm \frac{1}{2} )
( pm \frac{5}{8} )
( pm 8 )
( pm \frac{1}{4} )
( pm 5 )

Explanation:

1. By the Rational Root Theorem, rational roots' numerators are factors of the polynomial's constant term.
2. By the Rational Root Theorem, rational roots' denominators are factors of the polynomial's leading coefficient.
3. By the Remainder Theorem, the remainder of dividing a polynomial by $x-a$ is the polynomial's value at $x=a$. For $a=1$, the divisor is $x-1$.
4. Using the Rational Root Theorem for $8x^3 + 15x^2 -7x -5$, potential rational roots are $\pm\frac{\text{factors of 5}}{\text{factors of 8}}$. $\pm8$ has a numerator that is not a factor of the constant term 5, so it is not a potential root.

Answer:

1. constant term
2. leading coefficient
3. $x-1$
4. $\pm8$