Multiplying Functions and Degree Behavior
Multiplying polynomial functions (fg)(x) produces a new polynomial whose degree is the sum of the degrees of f and g. This concept is taught in Openstax Intermediate Algebra 2E, Chapter 5: Polynomials and Polynomial Functions. Knowing how degrees add under multiplication helps predict end behavior and classify the resulting polynomial.
Key Concepts
Property To find the product of two functions, $(fg)(x)$, multiply their polynomial expressions using the Distributive Property (or the vertical multiplication method). When you multiply two non zero polynomials, the degree (highest exponent) of the new combined function is exactly the sum of the degrees of the original functions.
Examples Binomial times Trinomial: Let $f(x) = x + 2$ and $g(x) = x^2 + 3x + 1$. Find $(fg)(x)$. Distribute the $x$: $x(x^2 + 3x + 1) = x^3 + 3x^2 + x$ Distribute the 2: $2(x^2 + 3x + 1) = 2x^2 + 6x + 2$ Combine like terms: $x^3 + 5x^2 + 7x + 2$ Predicting the Degree: Let $P(x) = 3x^5 + x$ (degree is 5) and $Q(x) = 2x^2 1$ (degree is 2). Without doing the full multiplication, we know the degree of the product $(PQ)(x)$ will be $5 + 2 = 7$.
Common Questions
How do you multiply two polynomial functions?
Multiply f(x) and g(x) using the distributive property, expanding all terms and then combining like terms.
What is the degree of the product of two polynomials?
The degree of (fg)(x) equals the sum of the degrees of f(x) and g(x).
Where is multiplying polynomial functions taught in Openstax?
This is in Openstax Intermediate Algebra 2E, Chapter 5: Polynomials and Polynomial Functions.
How does the leading coefficient of the product relate to those of the factors?
The leading coefficient of the product is the product of the leading coefficients of the two polynomials.
Why does the degree of the product equal the sum of degrees?
The highest-degree term in the product comes from multiplying the leading terms of each polynomial, and exponents add under multiplication.