Question

determine if the following statement is true or false. -1| 2 -2 3 -8 -2 4 -7 means (2x^3 - 2x^2 + 3x - 8)/(x + 1)=2x^2 - 4x + 7 - 15/(x + 1). 2 -4 7 -15 is the given statement true or false? false true

Explanation:

Step1: Recall synthetic - division rule

When we perform synthetic division of a polynomial \(P(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\) by \(x - c\), the result can be written as \(P(x)=(x - c)Q(x)+R\), where \(Q(x)\) is the quotient polynomial and \(R\) is the remainder. Here, we are dividing \(P(x)=2x^3-2x^2 + 3x-8\) by \(x+1\) (i.e., \(c=-1\)).

Step2: Analyze the synthetic - division result

The numbers in the synthetic - division setup:
The top - row numbers are the coefficients of the dividend polynomial \(2x^3-2x^2 + 3x-8\): \(2,-2,3,-8\). The number on the left is \(c=-1\).
The bottom - row numbers (after performing synthetic division) give the coefficients of the quotient polynomial and the remainder. The quotient polynomial \(Q(x)\) has coefficients \(2,-4,7\) (for \(2x^2-4x + 7\)) and the remainder \(R=-15\).
By the division algorithm for polynomials \(\frac{2x^3-2x^2 + 3x-8}{x + 1}=2x^2-4x + 7-\frac{15}{x + 1}\) since \(2x^3-2x^2 + 3x-8=(x + 1)(2x^2-4x + 7)-15\).

Answer:

true