Question

1. select the correct equation represented by the synthetic division problem

choose the best answer.
a shortcut method of dividing a polynomial by a linear polynomial by using only the coefficients of the terms of the polynomials is called __ division.
remainder
leading
constant
synthetic

choose the best answer.
the roots of a polynomial are values that make the polynomial equal __
zero
factors
real
constant

Explanation:

For Question 7:

Step1: Identify dividend & divisor

The coefficients $1, 1, -14, 6$ correspond to the polynomial $x^3 + x^2 -14x +6$. The divisor is $x-3$ (since we use $3$ in synthetic division for $x - k$ where $k=3$).

Step2: Match quotient from results

The bottom row $1, 4, -2, 0$ gives quotient $x^2 +4x -2$ and remainder $0$. So $\frac{x^3 + x^2 -14x +6}{x-3}=x^2 +4x -2$.

The described shortcut division method using only polynomial coefficients is the standard definition of this division type.

By definition, roots (or zeros) of a polynomial are input values that make the polynomial evaluate to 0.

Answer:

$\boldsymbol{(x^3 + x^2 -14x +6) \div (x - 3) = x^2 + 4x - 2}$

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For Question 8: