Math Problem Statement

What is (fg)(x)? f(x)= – 3x2–2x+1 g(x)=2x+4 Write your answer as a polynomial or a rational function in simplest form.

Solution

To find (fg)(x)(fg)(x), we need to multiply the functions f(x)f(x) and g(x)g(x). This is denoted as:

(fg)(x)=f(x)g(x)(fg)(x) = f(x) \cdot g(x)

Given the functions: f(x)=3x22x+1f(x) = -3x^2 - 2x + 1 and g(x)=2x+4g(x) = 2x + 4

We will multiply them:

(fg)(x)=(3x22x+1)(2x+4)(fg)(x) = (-3x^2 - 2x + 1)(2x + 4)

To simplify, we distribute each term of f(x)f(x) by each term of g(x)g(x):

  1. Multiply 3x2-3x^2 by each term in g(x)g(x): 3x22x=6x3-3x^2 \cdot 2x = -6x^3 3x24=12x2-3x^2 \cdot 4 = -12x^2

  2. Multiply 2x-2x by each term in g(x)g(x): 2x2x=4x2-2x \cdot 2x = -4x^2 2x4=8x-2x \cdot 4 = -8x

  3. Multiply 11 by each term in g(x)g(x): 12x=2x1 \cdot 2x = 2x 14=41 \cdot 4 = 4

Now, combine all the terms: (fg)(x)=6x312x24x28x+2x+4(fg)(x) = -6x^3 - 12x^2 - 4x^2 - 8x + 2x + 4

Combine like terms: (fg)(x)=6x316x26x+4(fg)(x) = -6x^3 - 16x^2 - 6x + 4

So, the simplified form of (fg)(x)(fg)(x) is:

(fg)(x)=6x316x26x+4(fg)(x) = -6x^3 - 16x^2 - 6x + 4


Would you like more details on any step or have further questions?

Here are 5 related questions:

  1. How do you factor polynomials such as 6x316x26x+4-6x^3 - 16x^2 - 6x + 4?
  2. What are the steps for finding the product of two polynomials in general?
  3. How do you check if a function is a factor of another polynomial?
  4. What is the meaning of the composition of functions (fg)(x)(f \circ g)(x)?
  5. How can you apply this to rational functions?

Tip: When multiplying polynomials, always distribute each term carefully and remember to combine like terms at the end for simplicity!

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Math Problem Analysis

Mathematical Concepts

Function Multiplication
Polynomial Multiplication
Algebra

Formulas

(fg)(x) = f(x) * g(x)

Theorems

Distributive Property

Suitable Grade Level

Grades 9-12