Question

true or false: according to the work below, ( x - 1 ) is a factor of ( -9x^4 + 10x^3 + 7x^2 - 6 ).

(the image shows a synthetic division setup with 1 as the divisor root, coefficients -9, 10, 7, 0, -6 in the first row, and the second row -9, 1, 8, 8, and the third row -9, 1, 8, 8, 2)

options:

• true
• false

Explanation:

Step1: Recall Factor Theorem

The Factor Theorem states that \(x - a\) is a factor of a polynomial \(f(x)\) if and only if \(f(a)=0\). When using synthetic division with root \(a\), the remainder is \(f(a)\).

Step2: Analyze Synthetic Division

In the given synthetic division, we are dividing by \(x - 1\) (so \(a = 1\)). The last number in the bottom row of synthetic division is the remainder. Here, the remainder is \(2\) (not \(0\)).

Step3: Apply Factor Theorem

Since the remainder when dividing \(-9x^{4}+10x^{3}+7x^{2}-6\) by \(x - 1\) is \(2\) (not \(0\)), by the Factor Theorem, \(x - 1\) is not a factor of the polynomial.

Answer:

False