Skill: Multiply polynomial functions

To multiply polynomial functions, multiply their outputs:

$$(fg)(x)=f(x)g(x).$$

Use the distributive property to multiply every term in one polynomial by every term in the other polynomial. Then combine like terms.

Suppose

$$f(x)=2x^2-3x+1 \qquad\text{and}\qquad g(x)=x-4.$$

We want to find $(fg)(x)$.

Step 1: Write the product

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

Step 2: Distribute each term

Multiply each term of $f(x)$ by both terms of $g(x)$:

$$\begin{aligned} (fg)(x) &=2x^2(x)+2x^2(-4)\\ &\quad+(-3x)(x)+(-3x)(-4)\\ &\quad+1(x)+1(-4) \end{aligned}$$

Step 3: Multiply the terms

When multiplying powers with the same base, add the exponents:

$$x^2\cdot x=x^{2+1}=x^3.$$

So,

$$(fg)(x)=2x^3-8x^2-3x^2+12x+x-4.$$

Step 4: Combine like terms

$$\boxed{(fg)(x)=2x^3-11x^2+13x-4}$$

The factored form is

$$(2x^2-3x+1)(x-4),$$

and the expanded form is

$$2x^3-11x^2+13x-4.$$

The product has degree $2+1=3$, and its leading coefficient is $2\cdot 1=2$. Remember that multiplying functions is different from composition: $(fg)(x)=f(x)g(x)$, while $f(g(x))$ means substitute $g(x)$ into $f(x)$.