Polynomials
A polynomial is an expression with variables and coefficients using addition, subtraction, and non-negative integer exponents.
Vocabulary
Degree: highest exponent. Leading coefficient: coefficient of the highest-degree term.
Monomial (1 term), Binomial (2 terms), Trinomial (3 terms).
Adding & Subtracting
Combine like terms — same variable and exponent.
(3x² + 2x − 1) + (x² − 5x + 4) = 4x² − 3x + 3
Multiplying (FOIL for Binomials)
FOIL: First, Outer, Inner, Last
(x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6
Special Products
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b² (difference of squares)
Factoring
GCF first, then factor the remaining trinomial or use difference of squares.
2x² − 8 = 2(x² − 4) = 2(x+2)(x−2)
Step by Step: Factoring a Trinomial
Factoring reverses multiplication. For x² + bx + c, you are hunting for two numbers that multiply to c and add to b.
Factor x² + 7x + 12
- List factor pairs of 12: 1×12, 2×6, 3×4.
- Which pair adds to 7?
3 + 4 = 7. - Write the factors:
(x + 3)(x + 4). - Check by FOIL:
x² + 4x + 3x + 12 = x² + 7x + 12. Correct.
Factor x² − 2x − 15
You need two numbers multiplying to −15 and adding to −2. A negative product means the signs differ. Try −5 and 3: the product is −15 and the sum is −2.
Answer: (x − 5)(x + 3)
Always Take the GCF First
Before anything else, pull out what every term shares. It makes the rest of the problem smaller, and forgetting it is the most common reason a factoring answer is marked incomplete.
Factor 3x² + 15x + 18
Every term is divisible by 3: 3(x² + 5x + 6)
Now factor what is left: 3(x + 2)(x + 3)
Degree and Shape
The degree of a polynomial predicts how its graph behaves, which is why teachers keep asking for it.
- Degree 1 (linear) graphs as a straight line.
- Degree 2 (quadratic) graphs as a parabola with one turn.
- Degree 3 (cubic) can turn twice and often has an S shape.
- In general, a polynomial of degree n has at most n roots and at most n − 1 turns.
Common Mistakes
- Combining unlike terms.
3x²and3xare not like terms and cannot be added. - Forgetting to distribute the minus sign when subtracting a polynomial.
(x² + 2x) − (x² − 3x)gives5x, not−x. - Claiming
(a + b)² = a² + b². The middle term2abis real; test it with numbers and you will see the gap. - Treating
a² + b²as a difference of squares. Only subtraction factors that way.
Where You Will Use This
- Area and volume formulas turn into polynomials when a dimension is written in terms of another.
- Physics uses polynomials to describe motion under constant acceleration.
- Economics models cost and revenue curves as polynomials.
- Computer graphics use polynomial curves to draw smooth shapes and fonts.
Summary
- A polynomial uses only whole-number exponents on its variables.
- Add and subtract by combining like terms; multiply with FOIL or full distribution.
- Memorize the special products — they turn up constantly.
- Factor by taking the GCF first, then finding two numbers with the right product and sum.
- Degree tells you the maximum number of roots and turns in the graph.
FAQ
Is 3/x a polynomial? No — exponents must be non-negative integers. Negative exponents disqualify it.
What is the difference between an expression and an equation? An expression like x² + 3x is something you simplify or factor. An equation has an equals sign and is something you solve.
How do I multiply polynomials bigger than binomials? FOIL is only a nickname for distributing. Multiply every term in the first by every term in the second, then combine like terms.
What if a trinomial will not factor? Some polynomials are prime over the integers. If no factor pair works, say so — or, if it is an equation, use the quadratic formula.
Why does factoring matter? Because a product equal to zero means one of the factors is zero. That single fact is how most polynomial equations get solved.
Quick Quiz
Test what you just learned. Choose the best answer for each question.