5.1 Add and Subtract Polynomials

Polynomial Basics

Polynomials are expressions built from variables, constants, and exponents combined with addition, subtraction, and multiplication. They show up constantly in later algebra topics, so getting comfortable adding and subtracting them now makes everything ahead smoother.

Degree of Polynomials

The degree of a polynomial is the highest exponent on its variable. It tells you the "power" of the polynomial and affects the shape of its graph.

• $3x^2 + 2x - 1$ has degree 2 because the highest exponent on $x$ is 2.
• A plain constant like $5$ or $-2$ has degree 0 (no variable means no exponent).

For polynomials with more than one variable, find the degree of each term by adding the exponents within that term, then take the highest sum.

• In $2x^2y + 3xy^2 - 4$, the term $2x^2y$ has exponents 2 and 1, which sum to 3. The term $3xy^2$ also sums to 3. The constant $-4$ has degree 0. So the polynomial's degree is 3.

Evaluating Polynomial Functions

To find the value of a polynomial function $f(x)$ at a specific input, substitute that number in for every $x$ and simplify.

Example 1: Find $f(2)$ when $f(x) = 2x^2 - 3x + 1$

1. Replace $x$ with 2: $f(2) = 2(2)^2 - 3(2) + 1$

2. Simplify: $f(2) = 2(4) - 6 + 1 = 8 - 6 + 1 = 3$

Example 2: Find $f(-2)$ when $f(x) = x^2 - 4x + 3$

1. Replace $x$ with $-2$: $f(-2) = (-2)^2 - 4(-2) + 3$

2. Simplify: $f(-2) = 4 + 8 + 3 = 15$

Watch the signs carefully when substituting negative values. Squaring a negative gives a positive: $(-2)^2 = 4$. And multiplying two negatives also gives a positive: $-4(-2) = +8$. Sign errors here are one of the most common mistakes.

Polynomial Operations

Adding and Subtracting Polynomials

Both operations come down to one key idea: combine like terms. Like terms have the same variable raised to the same exponent.

Adding polynomials is straightforward. Group the like terms and add their coefficients:

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

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

$= 3x^2 - x + 1$

Subtracting polynomials has one extra step: distribute the negative sign to every term in the second polynomial before combining. This is where mistakes happen most often.

$(5x^2 + x - 1) - (4x^2 - 2x + 3)$

$= (5x^2 + x - 1) + (-4x^2 + 2x - 3)$

$= (5x^2 - 4x^2) + (x + 2x) + (-1 - 3)$

$= x^2 + 3x - 4$

Notice that $-2x$ became $+2x$ and $+3$ became $-3$ after distributing the negative. Every sign in the second polynomial flips.

Combining Polynomial Functions

When you have two polynomial functions and need to add or subtract them, you're doing the same process as above, just written in function notation.

If $f(x) = 3x^2 - 2x + 1$ and $g(x) = 2x^2 + 4x - 3$:

• $(f + g)(x) = f(x) + g(x) = (3x^2 - 2x + 1) + (2x^2 + 4x - 3) = 5x^2 + 2x - 2$
• $(f - g)(x) = f(x) - g(x) = (3x^2 - 2x + 1) - (2x^2 + 4x - 3) = x^2 - 6x + 4$

The result is a new polynomial function. You can then evaluate it at any input just like you would any other function.

Properties of Polynomial Addition

Two familiar properties carry over to polynomials:

• Commutative property: Order doesn't matter when adding. $a + b = b + a$, so $(f + g)(x) = (g + f)(x)$.
• Associative property: Grouping doesn't matter when adding. $(a + b) + c = a + (b + c)$.

These properties apply to addition but not to subtraction. Changing the order or grouping in a subtraction problem will change the result, so be careful.