Multiply polynomial functions

Multiplying polynomial functions defines a new function by $(fg)(x)=f(x)g(x)$ ; its algebraic form is found by applying the distributive property to every term, adding exponents when multiplying like bases, and combining like terms. The learner connects expanded and factored forms, understands that the degree and leading coefficient of a product are determined by those of the factors (for nonzero polynomials), and distinguishes multiplication from function composition; abstract polynomial-ring operations and more advanced generalizations are outside this scope.

Detailed Explanation: 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 $f(x)$ .

Learn by doing: Multiply polynomial functions

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Algebraic Functions - Multiply Bracketed Terms, Same Variable (First Term)

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