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\(P(x)\)
\( + \ - \ \times \)

Understanding
Polynomials

Master how to add, subtract, and multiply polynomials and discover why they form a closed system under these operations.

Algebra 2

What Is a Polynomial?

A polynomial is an expression consisting of variables and coefficients, combined using only addition, subtraction, and multiplication with non-negative integer exponents.

Anatomy of a Polynomial

\( 5x^3 + 2x^2 - 4x + 9 \)
 Leading term Term Term Constant term
Degree = \(3\) (highest exponent) • \(4\) terms • Leading coefficient = \(5\)
Degree
Highest exponent of the variable
Term
Each product of a coefficient and variable power
Coefficient
The numerical factor of each term

Why Do Polynomials Matter?

Polynomials behave like integers: when you add, subtract, or multiply two of them, you always get another polynomial. This property is called closure.

1
Modeling: Polynomials model projectile motion, revenue curves, and growth patterns.
2
Foundation: They are the building blocks for rational expressions, calculus, and beyond.
3
Closure: Unlike division, adding, subtracting, and multiplying polynomials always yields a polynomial.

The Closure Property

Set of All Polynomials
\(P(x)\)
\(Q(x)\)
\( + \ - \ \times \)
 operation
\(R(x)\)
 still a polynomial! ✓ Closed under addition, subtraction, and multiplication

Adding Polynomials

To add polynomials, combine like terms: group terms that share the same variable and exponent, then add their coefficients.

1
Remove parentheses (addition does not change signs).
2
Identify like terms by matching variable and exponent.
3
Add the coefficients of each group of like terms.
4
Write in standard form (descending degree order).

Subtracting Polynomials

Subtracting is just like adding, but first you distribute the negative sign to every term in the polynomial being subtracted.

1
Distribute the negative to every term of the second polynomial.
2
Combine like terms just as with addition.
3
Simplify and write in standard form.

Multiplying Polynomials

Use the distributive property: multiply each term of the first polynomial by every term of the second, then combine like terms.

1
Distribute each term of the first polynomial across the second.
2
Multiply coefficients and add exponents for each pair.
3
Combine like terms and write in standard form.

Classifying Polynomials

Classification Number of Terms Example
Monomial 1 \( 7x^3 \)
Binomial 2 \( x^2 - 4 \)
Trinomial 3 \( 2x^2 + 3x + 1 \)
Polynomial 4+ \( x^4 - x^3 + 2x - 5 \)

Step-by-Step Approach

1
Identify the operation: Are you adding, subtracting, or multiplying?
2
Write each polynomial in standard form (descending degree).
3
For subtraction: Distribute the negative sign first.
4
For multiplication: Distribute each term, then multiply coefficients and add exponents.
5
Combine like terms and simplify.
6
Verify: Check the degree of the result. For addition/subtraction it should be at most the max degree of the inputs. For multiplication it should be the sum of the degrees.

Example: Addition

Find \( (3x^2 + 5x - 2) + (x^2 - 4x + 7) \)

Step 1: Group like terms
\((3x^2 + x^2) + (5x + (-4x)) + (-2 + 7)\)
 Step 2: Add coefficients
\(4x^2 + 1x + 5\)
 Result
\(4x^2 + x + 5\)

Example: Subtraction

Find \( (4x^3 - 2x + 6) - (x^3 + 5x^2 - 3) \)

Step 1: Distribute the negative
\(4x^3 - 2x + 6 - x^3 - 5x^2 + 3\)
 Step 2: Group like terms
\((4x^3 - x^3) + (-5x^2) + (-2x) + (6 + 3)\)
 Step 3: Combine
\(3x^3 - 5x^2 - 2x + 9\)
\(3x^3 - 5x^2 - 2x + 9\)

Example: Multiplication

Find \( (2x + 3)(x^2 - x + 4) \)

Distribute 2x:
\(2x(x^2) + 2x(-x) + 2x(4) = 2x^3 - 2x^2 + 8x\)
 Distribute 3:
\(3(x^2) + 3(-x) + 3(4) = 3x^2 - 3x + 12\)
 Combine like terms:
\(2x^3 + (-2x^2 + 3x^2) + (8x - 3x) + 12\)
\(2x^3 + x^2 + 5x + 12\)

FOIL: A Special Case

When multiplying two binomials, the distributive property simplifies to the FOIL pattern: First, Outer, Inner, Last.

\((a + b)\)
\((c + d)\)
\(F: ac\)
\(O: ad\)
\(I: bc\)
\(L: bd\)
\(= ac + ad + bc + bd\)

Common Mistakes

✕ Forgetting to Distribute the Negative
When subtracting, the negative must apply to every term, not just the first. \( -(x^2 + 3x) \neq -x^2 + 3x \).
✕ Adding Exponents When Adding Terms
When adding like terms, add coefficients only. Exponents stay the same: \( 3x^2 + 5x^2 = 8x^2 \), not \( 8x^4 \).
✕ Combining Unlike Terms
\( 4x^3 \) and \( 2x^2 \) are not like terms. They cannot be combined: \( 4x^3 + 2x^2 \neq 6x^5 \).

Operations Summary

Operation What to Do Degree of Result Closed?
Addition Combine like terms ≤ max(deg P, deg Q) ✓ Yes
Subtraction Distribute negative, combine like terms ≤ max(deg P, deg Q) ✓ Yes
Multiplication Distribute every term, add exponents deg P + deg Q ✓ Yes
Division Not always possible within polynomials May not be polynomial ✕ No

Question 1 of 5

What is \( (2x^2 + 3x - 1) + (x^2 - 5x + 4) \)?

Question 2 of 5

What is \( (5x^3 + 2x - 7) - (3x^3 - x + 4) \)?

Question 3 of 5

What is \( (x + 4)(x - 3) \)?

Question 4 of 5

Which of the following is not a polynomial?

Question 5 of 5

If \( P(x) \) has degree 3 and \( Q(x) \) has degree 2, what is the degree of \( P(x) \cdot Q(x) \)?