Classification, standard form, addition, subtraction, multiplication, FOIL, special products, geometric models, and real-world applications — the complete structural foundation for factoring and quadratics
In the chapters before this one, you worked with algebraic expressions involving one or two terms. You learned to evaluate expressions, combine like terms, apply the distributive property, and use exponent rules. All of that work prepared you for this chapter.
A polynomial is an organized algebraic expression built from terms with whole-number exponents. Polynomials appear throughout algebra, geometry, science, engineering, and finance. When you calculate the area of a rectangle with variable side lengths, write a formula for the height of a launched object, or model a growing quantity, you are often working with a polynomial.
This chapter introduces polynomials as a structured family of algebraic expressions. You will learn to classify them, identify their degree and standard form, and perform the four major operations: addition, subtraction, multiplication, and special products. By the end of the chapter, you will have the algebraic fluency needed for Chapter 11: Factoring, and eventually for Chapter 12: Quadratic Functions and Equations.
A landscaper is designing a rectangular garden. The length is described as (x + 5) feet and the width as (x + 3) feet, where x is an unknown base measurement. The landscaper wants to know: What is the area of the garden? If a second garden is added alongside the first with area 2x² + 7x square feet, what is the total combined area? How does the total area change as x grows? To answer these questions, you need to know how to multiply and add polynomial expressions. By the end of this chapter, you will be able to do exactly that.
Polynomials are among the most useful structures in all of algebra. Their importance comes from their versatility. A polynomial can describe a straight line, a curve, a path through the air, a financial model, or a geometric area. The operations you learn in this chapter — adding, subtracting, and multiplying polynomials — form the computational backbone for factoring in Chapter 11, quadratic equations in Chapter 12, and advanced algebra throughout Algebra II.
More immediately, this chapter will improve your control over algebraic expressions. Working carefully with polynomials trains you to track signs, manage exponents, organize terms, and think about algebraic structure. These are skills that carry into every area of mathematics.
An algebraic expression is a combination of numbers, variables, and operations. Not every algebraic expression is a polynomial. Polynomials are a specific, organized family of expressions with clear structural rules.
A polynomial is an algebraic expression consisting of one or more terms in which:
1. Variables have whole-number exponents (0, 1, 2, 3, …).
2. No variable appears in a denominator.
3. No variable appears under a radical (square root) sign.
4. No variable appears as an exponent.
3x² + 5x − 7
4x³ − 2x + 1
8
−6x
x⁴ + 2x³ − x² + 9x − 11
| Expression | Reason Not a Polynomial |
|---|---|
3/x + 5 | Variable in denominator |
√x + 2 | Variable under radical |
2ˣ + 1 | Variable as exponent |
x⁻² + 3 | Negative exponent on variable |
Polynomials can be added, subtracted, and multiplied — and the result is always another polynomial. That closure makes them exceptionally useful. When variables appear in denominators, as exponents, or under radicals, the expressions can behave very differently — with restrictions, asymptotes, or complex domains. Polynomials avoid all of that.
The subtraction signs in a polynomial belong to the terms that follow them. In 5x³ − 2x² + 7x − 4, the term −2x² has a coefficient of negative 2, not positive 2. Always carry the sign with the coefficient when identifying terms.
Classify each expression as a polynomial or not a polynomial. If not, explain why.
4x² − 3x + 15/x + 27x³ − x² + 83ˣ − 1−9x^(1/2) + 46x⁴ − x³ + 2x² − 5x + 10x⁻¹ + 3xPolynomials are classified by the number of terms they contain.
| Name | Number of Terms | Example |
|---|---|---|
| Monomial | 1 | 5x³ |
| Binomial | 2 | 3x + 7 |
| Trinomial | 3 | x² − 4x + 9 |
| Polynomial | 4 or more | 2x⁴ − x³ + 3x − 8 |
Like terms cannot make two separate terms. The expression 3x + 5x has two terms, but since they are like terms they simplify to 8x, which is a monomial. Always simplify before classifying.
5x⁴ − 3x + 7 b. −6x² c. x² + x − 12 d. 4x³ − 2x² + x − 9a. 5x⁴ − 3x + 7 → three terms → Trinomial
b. −6x² → one term → Monomial
c. x² + x − 12 → three terms → Trinomial
d. 4x³ − 2x² + x − 9 → four terms → Polynomial (four terms)
x² − 47x³2x² + 3x − 5x⁴ − 3x³ + x² − 7x + 2−128x + 1The degree of a polynomial describes how large its exponents are. Degree is one of the most important structural features of a polynomial because it determines the polynomial's shape when graphed and the way it behaves for large values of x.
The degree of a term is the sum of the exponents of all variables in that term. For single-variable polynomials — the focus in Algebra I — the degree of a term is simply the exponent on the variable. The degree of a polynomial is the highest degree among all its terms.
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | 7 |
| 1 | Linear | 3x + 1 |
| 2 | Quadratic | x² − 4x + 3 |
| 3 | Cubic | 2x³ + x |
| 4 | Quartic | x⁴ − 5 |
| 5 | Quintic | 3x⁵ + x³ |
A constant term like 7 has degree 0, not "no degree." You can think of it as 7x⁰, since x⁰ = 1.
6x³ − 2x + 5 b. 4x² − 9 c. −3 d. x⁴ − x³ + 2x² − 7x + 1a. Degrees: 3, 1, 0 → Degree 3 (Cubic)
b. Degrees: 2, 0 → Degree 2 (Quadratic)
c. One constant → Degree 0 (Constant)
d. Degrees: 4,3,2,1,0 → Degree 4 (Quartic)
Find the degree of each polynomial and give its special name if it has one.
5x − 3x² + 2x − 84x³ − x² + 2−11x⁴ + 73x⁵ − 2x³ + x − 6A polynomial is in standard form when its terms are written in descending order of degree: the highest-degree term comes first, and lower-degree terms follow in decreasing order. When a polynomial is in standard form, the leading term is the first term and the leading coefficient is its coefficient.
Standard form makes polynomials easier to compare, add, subtract, and factor. When you organize a polynomial consistently, like terms appear in predictable positions, which reduces errors.
5 + 3x² − x b. −4x + 7x³ + 2 − x²a. 3x² − x + 5 Degree: 2 Leading coefficient: 3
b. 7x³ − x² − 4x + 2 Degree: 3 Leading coefficient: 7
Negative signs belong to the term that follows them. When reordering, carry the sign with each term. 5 + 3x² − x has a term −x, not +x. Write it as 3x² − x + 5, not 3x² + x + 5.
Write in standard form. State the degree and leading coefficient.
3 + x² − 4x6x − 2x³ + 1−7 + 5x⁴ − x9x² + 2 − 3x³ + x4x − 10 + x⁵ − 2x³Adding polynomials is fundamentally the same as combining like terms. The key rule: only like terms can be combined. Like terms have identical variable parts — the same variables raised to the same exponents. When you add polynomials, you add the coefficients of matching terms and leave the variable parts unchanged.
You may add horizontally (grouping like terms in a single line) or vertically (aligning like terms in columns). Both methods produce the same result.
(3x² + 5x − 2) + (4x² − 3x + 7).(3x² + 5x − 2) + (4x² − 3x + 7)
= 3x² + 5x − 2 + 4x² − 3x + 7
= (3x² + 4x²) + (5x − 3x) + (−2 + 7)
= 7x² + 2x + 5
(2x³ − 4x + 1) + (5x³ + 3x² − 2x − 6). 2x³ + 0x² − 4x + 1
+ 5x³ + 3x² − 2x − 6
─────────────────────
7x³ + 3x² − 6x − 5
When one polynomial is missing a term that the other has, insert a zero placeholder such as 0x² to keep columns aligned in the vertical method. This prevents accidentally combining terms of different degrees.
Add. Write each answer in standard form.
(x² + 3x − 5) + (2x² − x + 4)(3x³ − x + 7) + (2x² + 4x − 3)(5x² − 4) + (−2x² + 3x + 9)(x⁴ + 3x² − 1) + (2x⁴ − x³ + x − 6)(6x − 5) + (3x + 2)(4x³ + 2x² − 7x + 1) + (−4x³ + x² + 5x − 3)Subtracting polynomials adds one layer of complexity: you must distribute the negative sign to every term of the polynomial being subtracted before combining like terms.
The rule: (A) − (B) = A + (−B). To subtract polynomial B from polynomial A, change every sign in B and then add.
(5x² − 3x + 2) − (2x² + 4x − 7).5x² − 3x + 2 − 2x² − 4x + 7
= (5x² − 2x²) + (−3x − 4x) + (2 + 7)
= 3x² − 7x + 9
(3x³ − x + 5) − (x³ + 2x² − 4x + 1).3x³ − x + 5 − x³ − 2x² + 4x − 1
= (3x³ − x³) + (−2x²) + (−x + 4x) + (5 − 1)
= 2x³ − 2x² + 3x + 4
The most common error in polynomial subtraction is failing to distribute the negative sign to every term. Students often correctly change the sign of the first term inside the parentheses but leave the remaining terms unchanged. Write out every sign change before you begin combining.
(4x² + x − 3) − (x² − 2x + 5)(7x³ − 2x + 4) − (3x³ + x² − x + 6)(6x² − 5) − (2x² + 3x − 1)(x⁴ + 2x² − 7) − (x⁴ + x³ − x² + 3)(9x − 4) − (5x − 10)2x² − 3x + 8 from 5x² + x − 2.Before multiplying polynomials with multiple terms, be fluent with multiplying monomials. The product rule from Chapter 9: when multiplying with the same base, add the exponents: xᵐ · xⁿ = xᵐ⁺ⁿ. Multiply the coefficients and add the exponents on the variable parts.
(3x²)(5x⁴) b. (−2x³)(4x) c. (6x²)(−3x⁵)a. (3 · 5)(x²⁺⁴) = 15x⁶
b. (−2 · 4)(x³⁺¹) = −8x⁴
c. (6 · (−3))(x²⁺⁵) = −18x⁷
(−5x³)(3x²)(2x).(−5 · 3 · 2)(x³⁺²⁺¹) = −30x⁶(4x³)(7x²)(−3x)(5x⁴)(2x⁵)(−6x³)(−4x²)(−3x²)(x³)(x⁵)(3x²)(4x)(2x³)(−2x⁴)(3x)(−x²)5 · (−2x³)To multiply a monomial by a polynomial with more than one term, apply the distributive property: multiply the monomial by each term of the polynomial separately.
a(b + c + d) = ab + ac + ad
This is the same distributive property from Chapter 1 — only now you are distributing a monomial with a variable and an exponent.
3x(x² − 4x + 5).3x · x² − 3x · 4x + 3x · 5
= 3x³ − 12x² + 15x
−2x²(4x³ − x + 7).−2x² · 4x³ −(−2x²) · x + (−2x²) · 7
= −8x⁵ + 2x³ − 14x²
When distributing a negative monomial, every term inside the parentheses changes sign. Students sometimes only change the first term. Work through every term in order and count them to make sure none was skipped.
2x(x + 5)4x²(3x − 2)−3x(x² + 4x − 1)5x³(2x² − x + 3)−x²(x³ − 5x + 8)7(2x² − 3x + 1)−4x²(x² − x − 6)3x³(x² + 2x − 4)Multiplying two binomials requires distributing each term of the first binomial across the entire second binomial. The result is a polynomial with up to four terms, which usually simplifies to a trinomial.
The formula: (a + b)(c + d) = ac + ad + bc + bd. Every term in the first binomial multiplies every term in the second, producing four products.
F — First terms: multiply the first terms of each binomial
O — Outer terms: multiply the outermost terms
I — Inner terms: multiply the innermost terms
L — Last terms: multiply the last terms of each binomial
FOIL is not a magical rule — it is a way to remember to multiply every pair of terms exactly once. Then combine any like terms.
(x + 3)(x + 5).F: x · x = x²
O: x · 5 = 5x
I: 3 · x = 3x
L: 3 · 5 = 15
x² + 5x + 3x + 15 = x² + 8x + 15
(x − 4)(x + 6).F: x · x = x²
O: x · 6 = 6x
I: −4 · x = −4x
L: −4 · 6 = −24
x² + 6x − 4x − 24 = x² + 2x − 24
(2x + 3)(3x − 5).F: 2x · 3x = 6x²
O: 2x · (−5) = −10x
I: 3 · 3x = 9x
L: 3 · (−5) = −15
6x² − 10x + 9x − 15 = 6x² − x − 15
Some students find an area model useful for multiplying binomials. Think of the product as the area of a rectangle, where each dimension is divided into two parts. For (x + 3)(x + 5), draw a rectangle with horizontal side split into x and 5, and vertical side split into x and 3. The four interior regions represent the four products.
| x | 5 | |
|---|---|---|
| x | x² | 5x |
| 3 | 3x | 15 |
Total area: x² + 5x + 3x + 15 = x² + 8x + 15. Each region is visible and no product is missed.
FOIL applies specifically to the product of two binomials. When multiplying a binomial by a trinomial, or a trinomial by a trinomial, you must extend the distribution more carefully. FOIL is a useful shortcut for the two-term case only.
(x + 2)(x + 7)(x + 5)(x − 3)(x − 4)(x − 6)(2x + 1)(x + 5)(3x − 2)(x + 4)(4x − 3)(2x − 5)(x + 8)(x − 8)(5x + 2)(3x − 1)(x + 1)(x + 1)(x − 7)(x + 2)Certain binomial products occur so frequently in algebra that their results are worth recognizing as patterns. Learning these special product formulas saves time, prevents errors, and — crucially — prepares you to reverse them in Chapter 11 when factoring.
(a + b)(a − b) = a² − b² Difference of Squares
(a + b)² = a² + 2ab + b² Square of a Sum
(a − b)² = a² − 2ab + b² Square of a Difference
When two binomials have the same first term and the same second term but with opposite signs, the product is a difference of two perfect squares. The middle terms cancel because −ab + ab = 0.
(x + 7)(x − 7).(x + 7)(x − 7) = x² − 7² = x² − 49(3x + 5)(3x − 5).(3x)² − 5² = 9x² − 25When you square a binomial sum, the result is a perfect square trinomial: the square of the first term, twice the product of the two terms, and the square of the second term. The middle term 2ab comes from ab + ab in the FOIL expansion.
(x + 4)².(x + 4)² = x² + 2(x)(4) + 4² = x² + 8x + 16(x − 6)².(x − 6)² = x² − 2(x)(6) + 6² = x² − 12x + 36(2x − 5)².(2x)² − 2(2x)(5) + 5² = 4x² − 20x + 25The single most common special-product error is writing (x + 4)² = x² + 16. This is wrong. Squaring a binomial means multiplying it by itself, not squaring each term separately. The middle term 2ab is always present.
(x + 4)² ≠ x² + 16
(x + 4)² = x² + 8x + 16 ← always include the middle term
Apply the appropriate special product formula.
(x + 9)(x − 9)(x + 3)²(x − 5)²(2x + 7)(2x − 7)(3x + 1)²(4x − 3)²(5x + 2)(5x − 2)(x − 10)²(6x + 5)²(x + 1)²Polynomials arise naturally in geometry whenever lengths, areas, and volumes involve variable measurements. When a rectangle has dimensions expressed as binomials, multiplying them produces a polynomial representing area. The same operations appear in perimeter formulas, bordered-region problems, and combined-shape situations.
(x + 7) and width (x + 2). Write a polynomial for its area.A = (x + 7)(x + 2)
= x² + 2x + 7x + 14
= x² + 9x + 14
(x + 5). A small square with side 3 is cut from one corner. Write a polynomial for the remaining area.Large square: (x + 5)² = x² + 10x + 25
Small square: 3² = 9
Remaining: x² + 10x + 25 − 9 = x² + 10x + 16
(3x + 4) and width (x − 1). Write polynomials for its perimeter and area.Perimeter: 2(3x + 4) + 2(x − 1) = 6x + 8 + 2x − 2 = 8x + 6
Area: (3x + 4)(x − 1)
= 3x² − 3x + 4x − 4
= 3x² + x − 4
(x + 6) and width (x + 1). Write a polynomial for its area.(2x + 3). Write a polynomial for its area.(4x + 5) and width (x − 2). Write polynomials for both perimeter and area.(2x + 4) and height x. Write a polynomial for its area. Recall A = ½bh.(x + 4). A path of width 1 foot surrounds it. Write a polynomial for the area of the path alone.Polynomials can describe many real-world quantities — not only geometric shapes, but also revenue, profit, projectile height, population growth, and more. The key skill is translating a verbal description into a polynomial expression and interpreting what the polynomial tells you about the situation.
(200 − 5p), where p is the price per ticket. Write a polynomial expression for total revenue.R = p(200 − 5p) = 200p − 5p²(x + 8) by (x + 4) feet. Room 2: (x + 3) by (x + 2) feet. Write a polynomial for total floor area.Room 1: (x + 8)(x + 4) = x² + 12x + 32
Room 2: (x + 3)(x + 2) = x² + 5x + 6
Total: 2x² + 17x + 38
P(x) = −x² + 80x − 1200, where x = units produced. Find profit at x = 40.P(40) = −(40)² + 80(40) − 1200
= −1600 + 3200 − 1200
= 400
(x + 10) and width (x + 6). Write a polynomial for its area. What is the area when x = 5?R = n(50 − n), where n is units sold. Expand this expression.h(t) = −16t² + 64t + 5. What is its height at t = 2 seconds?(x + 4). Write a polynomial for their combined area.P(x) = −2x² + 120x − 800. Find profit when x = 30.Correct: 3x² + 5x − 2 − x² + 4x − 1 = 2x² + 9x − 3
Every sign inside the parentheses changes when subtracting.
Correct: (x + 5)² = x² + 10x + 25
Squaring a binomial means multiplying it by itself. Always write (x + 5)(x + 5) and FOIL.
Correct: 3x² + 4x² = 7x²
When adding like terms, add coefficients only. The exponent stays the same.
Exponents are only added when multiplying (Product Rule), not when adding.
Correct: 5x² + 3x cannot be simplified further.
x² and x are different degrees — they are unlike terms and cannot be combined.
Correct: 3x(x² + 4x − 2) = 3x³ + 12x² − 6x
Count the terms inside before distributing. Verify your result has that many terms.
Correct: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
FOIL requires all four products. The Outer and Inner terms (5x and 3x) were skipped.
Correct: The degree is 2 (quadratic). Degree is determined by the highest exponent,
not by the presence of any particular term.
Each problem contains an error. Find and correct it.
(2x² − 5x + 3) − (x² + 4x − 1) = x² − x + 4. Check each step. Is there an error? If so, where and what is the correct answer?(x + 6)² = x² + 36. What did the student do wrong? What is the correct answer?4x³ + 2x³ = 6x⁶. Identify the error and give the correct sum.5x²(x − 3) = 5x³ − 3. Find the error and give the correct product.(x + 4)(x − 9) = x² − 36. Check by FOIL and identify what was done wrong.2 + 3x − 4x³ + x²? A student says "degree 1 because it starts with the constant 2." Explain the error.You have now learned how to build polynomials by adding, subtracting, and multiplying. In Chapter 11, you will learn to factor polynomials — to work in the reverse direction.
Every factoring problem is a multiplication problem read backward. You know that (x + 3)(x + 5) = x² + 8x + 15. Factoring asks the reverse: given x² + 8x + 15, can you find the two binomials whose product it equals? You will find (x + 4)(x + 5) — and verify it by multiplying back.
Factoring is one of the most powerful techniques in algebra. It allows you to rewrite complex expressions in simpler forms, solve polynomial equations by finding where a polynomial equals zero, and understand the roots and graphs of functions. In Chapter 12, you will use factoring to solve quadratic equations — to find the exact values of x where a parabola crosses the x-axis.
To succeed in Chapter 11, be confident with: identifying greatest common factors from lists of integers; multiplying binomials using FOIL and the area model; recognizing the special product patterns; and combining like terms carefully with correct signs. If any of these feel uncertain, return to Sections 10.5 through 10.10 for review.
Chapter 11 will teach four major strategies: (1) Greatest common factor, (2) Factoring trinomials with leading coefficient 1, (3) Factoring trinomials with leading coefficient greater than 1, and (4) Difference of squares and perfect square trinomial patterns. Each builds directly on what you practiced in this chapter.
The special products you learned in Section 10.10 will reappear in factored form in Chapter 11. Learning to recognize a² − b² and perfect square trinomials from the product side now will help you recognize and factor them on sight later.
(x + 4)(x + 5). Now try to reverse the process: given x² + 9x + 20, can you identify the two binomials?(x − 3)(x + 3). What pattern do you see in the result? Why do the middle terms cancel?(x + 7)². Now given x² + 14x + 49, can you identify the original binomial square?6x³ + 9x² − 12x? Factor it out.2x(x + 4). Now given 2x² + 8x, can you reverse the process?Classify each as monomial, binomial, trinomial, or polynomial with four+ terms.
x³ − 54x⁴ − x³ + 2x² − x + 7−3x² + 7x − 118xx² − 4State the degree and leading coefficient of each.
5x³ − 2x + 1−x⁴ + 6x² − 373x² − x−2x⁵ + x³ − 9x(3x² + x − 5) + (2x² − 3x + 8)(4x³ − 2x + 7) − (x³ + 3x² − x + 4)(x² − 9) + (3x² + 5x + 2)(6x⁴ + x² − 3) − (4x⁴ − x³ + 2x² − 1)(5x − 4) + (3x² − 2x + 9) − (x² + x − 6)(2x³)(5x⁴)(−3x²)(7x³)4x(3x² − 5x + 2)−2x³(x² − 4x + 1)(x + 4)(x + 7)(x − 3)(x + 8)(2x + 5)(3x − 2)(4x − 1)(x + 6)(x + 3)(x² − 2x + 5)(2x − 1)(x² + 3x − 4)(x + 8)(x − 8)(x + 7)²(x − 4)²(3x + 2)(3x − 2)(2x − 5)²(4x + 1)²(5x − 3)(5x + 3)(x + 10)²(3x − 7)²(6x + 1)(6x − 1)Define variables where needed, write expressions, and answer the specific question.
(x + 9) and width (x + 3). Write a polynomial for its area.(2x + 5). Write a polynomial for its area.(3x + 4) feet long and (x + 2) feet wide. Write polynomials for the perimeter and area of the pool.R = n(80 − 2n), where n is units sold. Expand and find revenue when n = 15.(x + 6) by (x + 4) feet. The garden itself measures x by (x + 2) feet. Write a polynomial for the area of the path alone.h(t) = −16t² + 48t + 6. Find the height at t = 1 and t = 2.(x + 5)(x + 3) and (x + 4)(x + 1). Write and simplify a polynomial for their combined area.Each problem below contains an error. Find and correct it.
What did the student do wrong? Write the correct expansion.
Identify the error and find the correct answer.
Is this correct? Explain your answer.
What is the correct sum? Explain why the exponents should not be added.
What did the student skip? Apply FOIL correctly and write the right answer.
7 − 4x + x³ − 2x² in standard form. State the degree and leading coefficient.−6x² as a monomial, binomial, or trinomial.3x⁴ − x² + 8? What is this polynomial called?(2x² + 5x − 3) + (x² − 4x + 7)(6x³ − 2x + 1) − (3x³ + x² − 4x + 5)(3x²)(−4x⁵)2x(x² − 5x + 3)(x + 4)(x − 9)(3x + 2)(2x − 7)(x + 11)(x − 11)(x + 6)²(2x − 3)²(x + 5) and width (x + 2). Write a polynomial for its area.P(x) = −3x² + 90x − 600. Find profit when x = 20.(x − 7)² = x² − 49.Everything you practiced in this chapter — multiplying monomials, distributing, applying FOIL, and recognizing special product patterns — has prepared you to work in reverse.
Chapter 11: Factoring begins with the same polynomials you have been building and asks you to disassemble them. Given x² + 10x + 24, you will find (x + 4)(x + 6). Given 9x² − 25, you will recognize the difference of squares and factor it as (3x + 5)(3x − 5). Given 4x³ − 8x² + 12x, you will factor out the greatest common monomial factor of 4x.
After Chapter 11, Chapter 12 will use factoring as a primary tool for solving quadratic equations. When you solve x² + 8x + 15 = 0 by writing (x + 3)(x + 5) = 0 and finding x = −3 or x = −5, you will be using the skills from both this chapter and the next together.
The progression is: multiply → factor → solve → model. You have completed the first step.