Question

1. if $f(x)\cdot g(x)=6x^2 - 29x + 35$ and $f(x)=(2x - 5)$, find $g(x)$.
2. if $g(x)=(x - 3)$ and $f(x)\cdot g(x)=2x^2 - 25x + 57$, find $f(x)$.

Explanation:

Problem 1

Step1: Recall division of polynomials

Given \( f(x) \cdot g(x) = 6x^2 - 29x + 35 \) and \( f(x) = 2x - 5 \), we need to find \( g(x) \) by dividing \( 6x^2 - 29x + 35 \) by \( 2x - 5 \).

Step2: Perform polynomial long division

Divide \( 6x^2 - 29x + 35 \) by \( 2x - 5 \).
First term: \( \frac{6x^2}{2x} = 3x \). Multiply \( 2x - 5 \) by \( 3x \): \( 6x^2 - 15x \).
Subtract from \( 6x^2 - 29x + 35 \): \( (6x^2 - 29x + 35) - (6x^2 - 15x) = -14x + 35 \).
Next term: \( \frac{-14x}{2x} = -7 \). Multiply \( 2x - 5 \) by \( -7 \): \( -14x + 35 \).
Subtract: \( (-14x + 35) - (-14x + 35) = 0 \). So \( g(x) = 3x - 7 \).

Step1: Recall division of polynomials

Given \( f(x) \cdot g(x) = 2x^2 - 25x + 57 \) and \( g(x) = x - 3 \), we need to find \( f(x) \) by dividing \( 2x^2 - 25x + 57 \) by \( x - 3 \).

Step2: Perform polynomial long division

Divide \( 2x^2 - 25x + 57 \) by \( x - 3 \).
First term: \( \frac{2x^2}{x} = 2x \). Multiply \( x - 3 \) by \( 2x \): \( 2x^2 - 6x \).
Subtract from \( 2x^2 - 25x + 57 \): \( (2x^2 - 25x + 57) - (2x^2 - 6x) = -19x + 57 \).
Next term: \( \frac{-19x}{x} = -19 \). Multiply \( x - 3 \) by \( -19 \): \( -19x + 57 \).
Subtract: \( (-19x + 57) - (-19x + 57) = 0 \). So \( f(x) = 2x - 19 \).

Answer:

\( g(x) = 3x - 7 \)

Problem 2