Lesson 1: Polynomial Functions

New Concept

Polynomials build on functions you already know, like linear and quadratic ones. They are expressions like $f(x) = a_n x^n + \dots + a_0$, which are powerful tools for modeling real-world scenarios from economics to engineering.

What’s next

You'll apply this definition by multiplying and factoring polynomials through interactive examples and practice problems. Let's start building your skills.

Property

A polynomial function has the form

where $a_0, a_1, a_2, \ldots, a_n$ are constants and $a_n \neq 0$. The coefficient $a_n$ of the highest power term is called the lead coefficient. Polynomials can be written in descending powers, where terms are ordered from the highest degree to the lowest, or in ascending powers, where terms are ordered from lowest degree to highest.

Examples

• The expression $p(x) = 7x^4 - 3x^2 + 5$ is a polynomial. Its degree is 4 and its lead coefficient is 7.
• The polynomial $q(x) = 5x - 2x^3 + 8$ written in descending powers is $q(x) = -2x^3 + 5x + 8$.
• The polynomial $r(x) = 4x^3 + x^5 - 9$ written in ascending powers is $r(x) = -9 + 4x^3 + x^5$.

Explanation

A polynomial is an expression built from variables and coefficients, using only addition, subtraction, multiplication, and non-negative integer exponents. The lead coefficient is simply the number in front of the term with the biggest exponent.

Property

The degree of a product of non-zero polynomials is the sum of the degrees of the factors. That is: If $P(x)$ has degree $m$ and $Q(x)$ has degree $n$, then their product $P(x)Q(x)$ has degree $m + n$.

Examples

• If you multiply a polynomial of degree 3 by a polynomial of degree 4, the resulting polynomial will have a degree of $3 + 4 = 7$.
• Let $P(x) = 3x^5 + x$ and $Q(x) = 2x^2 - 1$. The degree of $P(x)Q(x)$ is $5 + 2 = 7$.
• The lead term of the product $(4x^2 + 3x)(2x^3 - 5)$ is found by multiplying the lead terms of each factor: $(4x^2)(2x^3) = 8x^5$. The degree is 5.

Explanation

When you multiply polynomials, their degrees add up. To find the degree of the resulting polynomial, just sum the degrees of the original polynomials you are multiplying together. This is a quick way to know the final outcome without full calculation.

Property

1. $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$
2. $(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$

Examples

• To expand $(a + 4)^3$, use the first formula with $x=a$ and $y=4$: $a^3 + 3(a)^2(4) + 3(a)(4)^2 + 4^3 = a^3 + 12a^2 + 48a + 64$.
• To expand $(2b - 1)^3$, use the second formula with $x=2b$ and $y=1$: $(2b)^3 - 3(2b)^2(1) + 3(2b)(1)^2 - 1^3 = 8b^3 - 12b^2 + 6b - 1$.
• To expand $(z^2 + 3)^3$, use the first formula with $x=z^2$ and $y=3$: $(z^2)^3 + 3(z^2)^2(3) + 3(z^2)(3)^2 + 3^3 = z^6 + 9z^4 + 27z^2 + 27$.

Explanation

These formulas are shortcuts for expanding expressions like $(a+b)^3$ without performing a lengthy multiplication. Memorizing these patterns for the sum and difference of two cubed terms saves time and helps prevent common algebra mistakes.

Property

1. $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$
2. $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

Examples

• To factor $a^3 + 125$, identify $x=a$ and $y=5$. Using the first formula, we get $(a + 5)(a^2 - 5a + 25)$.
• To factor $64b^3 - 1$, identify $x=4b$ and $y=1$. Using the second formula, we get $(4b - 1)(16b^2 + 4b + 1)$.
• To factor $m^6 - 8n^3$, identify $x=m^2$ and $y=2n$. Using the second formula, we get $(m^2 - 2n)(m^4 + 2m^2n + 4n^2)$.

Explanation

These special formulas help you factor expressions that are a sum or difference of two perfect cubes. Recognizing these patterns is a key skill for simplifying complex polynomials and solving cubic equations by breaking them into smaller parts.