Q5.69 - Intermediate Algebra

Question

For function $$\begin{gathered} f\left(x\right)=x-5 and g\left(x\right)=x^2-2x+3 , \\ Find \left(a\right) (f.g)\left(x\right) \\ \left(b\right) (f.g)\left(2\right) \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right)Multiply the given binomial function \\ (f.g)\left(x\right)=(x-5)(x^2-2x+3) \\ =x^3-7x^2+13x-15 \\ \left(b\right) Multiply the given binomial function \\ (f.g)\left(2\right)=(x-5)(x^2-2x+3) \\ =(2-5)(2^2-2\times 2+3) \\ =-9 \end{gathered}$$

Step 1

Given information:

$f\left(x\right)=x-5 and g\left(x\right)=x^2-2x+3$

Step 2

Multiplying polynomials require only three steps.

• First, multiply each term in one polynomial by each term in the other polynomial using the distributive law.
• Add the powers of the same variables using the exponent rule.
• Then, simplify the resulting polynomial by adding or subtracting the like terms.

Step 3 Part(a) Step1. Multiply the Polynomials

$$\begin{gathered} part\left(a\right) Step 1.Multiply the polynomials \\ \left(a\right) \\ (f.g)\left(x\right)=f\left(x\right).g\left(x\right) \\ Subtitute for f\left(x\right) and g\left(x\right) (f.g)\left(x\right)=(x-5)(x^2-2x+3) \\ Multiply the polynomials (f.g)\left(x\right)=x(x^2-2x+5)-5(x^2-2x+3) \\ Distribute (f.g)\left(x\right)=x^3-2x^2+5x-5x^2+10x-15 \\ Combine like terms (f.g)\left(x\right)=x^3-7x^2+15x-15 \end{gathered}$$

Part(b) Step1. Multiply the Polynomials

$$\begin{gathered} \left(b\right) \\ In part \left(a\right) we found (f.g)\left(x\right) and now asked to find(f.g)\left(2\right) \\ (f.g)\left(x\right)=x^3-7x^2+15x-15 \\ to find (f.g)\left(2\right) subtitute x=2 (f.g)\left(2\right)={\left(2\right)}^3-7{\left(2\right)}^2+15\left(2\right)-15 \\ (f.g)\left(2\right)=8-28+30-15 \\ (f.g)\left(2\right)=-5 \end{gathered}$$