Question

question 1 of 5
select the correct answer.
one factor of this polynomial is (x - 7).
x³ - 4x² - 22x + 7
using synthetic division, what is the other factor of the polynomial?
x³ + 3x² - x
x³ + 3x² + x
x² + 3x + 1
x² + 3x - 1

Explanation:

Step1: Set up synthetic division

The root for the factor \((x - 7)\) is \(x = 7\). The coefficients of the polynomial \(x^{3}-4x^{2}-22x + 7\) are \(1\), \(-4\), \(-22\), and \(7\). We set up synthetic division as follows:

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Step2: Interpret the result

The numbers at the bottom row (excluding the last zero, which is the remainder) are the coefficients of the quotient polynomial. The quotient polynomial has degree one less than the original polynomial (since we divided a cubic by a linear factor), so it is a quadratic polynomial. The coefficients are \(1\) (for \(x^{2}\)), \(3\) (for \(x\)), and \(-1\) (constant term). So the quotient polynomial is \(x^{2}+3x - 1\).

Answer:

\(x^{2}+3x - 1\) (which corresponds to the option \(x^{2}+3x - 1\))