Polynomials An Introduction

Introduction

Polynomials: An Introduction is designed for learners in Grades 7–8. You will learn how to recognize polynomials, name their parts, simplify them by combining like terms, evaluate them for given values, and perform basic addition, subtraction, and multiplication. You will also connect symbolic expressions with tables, area models, and graphs.

A polynomial is a finite algebraic expression built from numbers and variables using addition, subtraction, and multiplication, where variable exponents are nonnegative whole numbers. In this course, we mainly work with one variable, usually $x$ . A typical example is $4 x^{3} - 2 x^{2} + 7 x - 5$ .

Polynomials are useful because they describe patterns. They can represent areas, costs, motion, sequences, and graphs. Learning to read and manipulate them gives you a strong foundation for later work in Algebra, quadratic equations, functions, and mathematical modeling.

After watching, identify three words that seem important. Keep them in mind as you read the vocabulary section below.

Building Blocks of a Polynomial

Terms, coefficients, variables, and exponents

Consider the polynomial $5 x^{3} - 2 x + 9$ . It has three terms: $5 x^{3}$ , $- 2 x$ , and $9$ . Terms are separated by addition or subtraction signs.

In $5 x^{3}$ , the number $5$ is the coefficient, $x$ is the variable, and $3$ is the exponent. The term $9$ has no variable, so it is a constant term.

Word	Meaning	Example in $6 x^{2} - 3 x + 8$
Term	One part of an expression separated by addition or subtraction	$6 x^{2}$
Coefficient	A numerical factor multiplying a variable part	$6$
Variable	A symbol that can stand for different numbers	$x$
Exponent	A number showing repeated multiplication of a base	$2$
Constant term	A term with no variable	$8$

Monomials, binomials, and trinomials

A monomial has one term, such as $7 x^{2}$ . A binomial has two terms, such as $x + 4$ . A trinomial has three terms, such as $x^{2} + 5 x + 6$ . A polynomial can have one, two, three, or more terms.

The names help you describe structure, but the important idea is that every term follows the exponent rule for polynomials.

What counts as a polynomial?

For the school-level polynomials in this course, exponents on variables must be $0, 1, 2, 3, \dots$ . Expressions such as $3 x^{2} - 4 x + 1$ and $- 7 x^{5} + 2$ are polynomials. The expression $2 / x + 3$ is not a polynomial in $x$ because $2 / x = 2 x^{- 1}$ . The expression $\sqrt{x} + 1$ is not a polynomial in $x$ because $\sqrt{x} = x^{1 / 2}$ .

A constant such as $12$ is also a polynomial. Its degree is $0$ , as long as the constant is not zero.

Degree and Standard Form

Finding the degree

The degree of a nonzero polynomial in one variable is the greatest exponent of the variable after like terms have been combined. For example, $7 x^{4} - 3 x^{2} + x - 6$ has degree $4$ .

The coefficient of the highest-degree term is called the leading coefficient. In $7 x^{4} - 3 x^{2} + x - 6$ , the leading coefficient is $7$ .

Standard form usually lists terms from greatest exponent to least exponent. Thus $4 - 2 x^{3} + 5 x$ becomes $- 2 x^{3} + 5 x + 4$ .

The graph above represents a degree-2 polynomial. A degree-2 polynomial function is called a quadratic function, and its graph is a parabola.

The graph above represents a degree-3 polynomial. You do not need to predict every feature of cubic graphs yet; the main point is that polynomial expressions can define functions whose shapes can be studied.

Like Terms and Simplifying

Recognizing like terms

Like terms have exactly the same variable part with exactly the same exponents. For example, $3 x^{2}$ and $- 5 x^{2}$ are like terms. The terms $3 x^{2}$ and $3 x$ are not like terms because their exponents differ.

To combine like terms, add or subtract only their coefficients and keep the common variable part. For example:

$4 x^{2} + 7 x^{2} = 11 x^{2}$

$9 x - 3 x + 5 = 6 x + 5$

$2 x^{2} + 3 x - 5 + x^{2} - 4 x + 8 = 3 x^{2} - x + 3$

A useful self-check is to ask: Did I combine only terms with matching variables and exponents?

Evaluating Polynomials

To evaluate a polynomial, replace the variable with a given number and calculate. Use parentheses around a substituted negative number so that signs remain clear.

For $P (x) = 2 x^{2} - 3 x + 4$ and $x = 3$ :

$P (3) = 2 (3)^{2} - 3 (3) + 4 = 18 - 9 + 4 = 13$

For $x = - 2$ :

$P (- 2) = 2 (- 2)^{2} - 3 (- 2) + 4 = 8 + 6 + 4 = 18$

Evaluation connects algebraic expressions to numerical outputs. This is one reason polynomials can be used as functions.

Adding and Subtracting Polynomials

Adding

To add polynomials, remove grouping symbols and combine like terms.

$(3 x^{2} + 2 x - 5) + (x^{2} - 7 x + 4)$

$= 3 x^{2} + x^{2} + 2 x - 7 x - 5 + 4$

$= 4 x^{2} - 5 x - 1$

Subtracting

When subtracting a polynomial, distribute the negative sign to every term in the polynomial being subtracted.

$(5 x^{2} + x - 3) - (2 x^{2} - 4 x + 6)$

$= 5 x^{2} + x - 3 - 2 x^{2} + 4 x - 6$

$= 3 x^{2} + 5 x - 9$

A common mistake is changing the sign of only the first term after a subtraction sign. Check every term before combining like terms.

Multiplying Polynomials

A monomial times a polynomial

Use the distributive property to multiply the monomial by every term.

$3 x (2 x^{2} - 5 x + 4) = 6 x^{3} - 15 x^{2} + 12 x$

When multiplying powers with the same base, add their exponents. For example, $x^{2} \cdot x^{3} = x^{5}$ .

A binomial times a binomial

You can use an area model or repeated distribution.

$(x + 3) (x + 2)$

$= x (x + 2) + 3 (x + 2)$

$= x^{2} + 2 x + 3 x + 6$

$= x^{2} + 5 x + 6$

The image shows how coefficients in Pascal's triangle connect to binomial expansion. This is an extension: you do not need the general formula yet, but you can look for patterns in expansions such as $(x + 1)^{2}$ and $(x + 1)^{3}$ .

Polynomials as Models and Graphs

Polynomials are more than symbol puzzles. They can describe changing quantities.

Suppose a rectangle has side lengths $x + 2$ and $x + 5$ . Its area is

$(x + 2) (x + 5) = x^{2} + 7 x + 10$ .

This polynomial tells you the area for any allowed value of $x$ . For example, if $x = 3$ , the side lengths are $5$ and $8$ , and the area is $40$ . Evaluating the polynomial gives the same result: $3^{2} + 7 (3) + 10 = 40$ .

Polynomial functions also have graphs. The places where a graph crosses the x-axis correspond to inputs that make the polynomial equal to zero.

In the graph above, the x-intercepts are also called roots or zeros of the quadratic polynomial. For Grades 7–8, focus on the connection: an algebraic rule can generate numerical values, a table, and a graph.

A Strategy for Solving Polynomial Problems

When you meet a polynomial problem, first identify what the question asks. Then check the terms and exponents, rewrite in standard form if useful, and decide whether you need to simplify, evaluate, add, subtract, or multiply. Keep like terms aligned and use the distributive property carefully.

A strong final check uses structure. If you add two degree-2 polynomials, the result cannot suddenly contain an $x^{5}$ term. If you multiply $x^{2}$ by $x^{3}$ , the product should contain $x^{5}$ , not $x^{6}$ . Estimating the form of an answer can catch errors before you move on.

Interactive Tasks

Quiz: Test Your Knowledge

Which expression is a polynomial in x? (3x^2 - 5x + 7) (!2 divided by x + 1) (!square root of x + 4) (!x to the negative 2 + 3)

What is the coefficient of x^3 in 6x^3 - 2x + 9? (6) (!3) (!2) (!9)

What is the degree of 4x^5 - x^2 + 8? (5) (!4) (!2) (!8)

What is the simplified form of 7x^2 - 3x^2 + 2x? (4x^2 + 2x) (!4x + 2x) (!10x^2 + 2x) (!4x^3 + 2x)

What is the value of 2x^2 + 1 when x is 3? (19) (!13) (!7) (!37)

What is the sum of 3x^2 + 4x and x^2 - 2x? (4x^2 + 2x) (!4x^2 + 6x) (!3x^2 + 2x) (!4x + 2)

What is the result of subtracting x^2 + 3x from 4x^2 + 5x? (3x^2 + 2x) (!5x^2 + 8x) (!3x^2 + 8x) (!5x^2 + 2x)

What is 2x times 3x^2 - 4? (6x^3 - 8x) (!6x^2 - 8) (!5x^3 - 4x) (!6x^3 - 4)

Which expression is written in standard form? (5x^3 - 2x^2 + x - 7) (!x - 7 + 5x^3 - 2x^2) (!-7 + x - 2x^2 + 5x^3) (!x + 5x^3 - 7 - 2x^2)

What does an x-intercept of a polynomial graph represent? (An input where the polynomial equals zero) (!The largest coefficient) (!The degree of the polynomial) (!The constant term only)

Memory Game

Polynomial	A finite algebraic expression with nonnegative whole-number exponents on its variables
Coefficient	A number multiplying a variable part
Variable	A symbol that can represent different numerical values
Degree	The greatest exponent in a nonzero one-variable expression after simplification
Constant	A term that contains no variable

Drag and Drop

Match the correct terms.	Topic
Same variable and exponent	Like terms
Greatest exponent	Degree
Number multiplying a variable	Coefficient
Replace a variable with a number	Evaluate
Multiply across a sum	Distributive property

...

Crossword Puzzle

Monomial	What do you call a polynomial with one term?
Coefficient	What is the numerical factor multiplying a variable part?
Variable	What symbol can stand for different values?
Degree	What name is given to the greatest exponent of a nonzero polynomial?
Constant	What is a term with no variable called?
Exponent	What tells how many times a base is used as a factor?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Polynomial vocabulary card: Create a one-page illustrated vocabulary card showing term, coefficient, variable, exponent, constant, and degree with your own examples.
Polynomial sorting challenge: Make twelve expression cards, including polynomials and non-polynomials, then sort them and explain every decision.
Like-term color map: Write a polynomial with at least eight terms and design a visual system that shows which terms can be combined.
Evaluation table: Choose one quadratic polynomial and create a table of its values for at least seven different integer inputs.

Standard

Rectangle area model: Design a rectangle with binomial side lengths, draw an area model, and show how the expanded polynomial represents the total area.
Polynomial interview: Interview a classmate about which step in simplifying polynomials is hardest, then write a short explanation that addresses the difficulty.
Graph and rule comparison: Use graphing technology or graph paper to compare a linear polynomial and a quadratic polynomial, then describe at least three differences.
Error analysis poster: Invent three believable mistakes in polynomial addition, subtraction, or multiplication and create a poster that corrects each one.

Advanced

Polynomial modeling project: Build a polynomial model for a geometric or numerical pattern, explain what each term means, and test the model with several values.
Binomial pattern investigation: Expand several powers of x plus 1, compare the coefficients with Pascal's triangle, and write a conjecture about the pattern.
Polynomial tutorial video: Produce a two- to four-minute teaching video that explains one polynomial operation using a worked example and a self-check.
Multiple representations investigation: Start with one quadratic polynomial and connect its symbolic form, value table, area interpretation when possible, and graph in a short report.

Learning Assessment

Structure and classification: Classify a set of unfamiliar expressions as polynomials or non-polynomials and justify each choice using the exponent and operation rules.
Reasoning with errors: Analyze a worked subtraction problem containing a sign error, explain why the error changes the result, and repair the solution.
Representation transfer: Given a rectangle with algebraic side lengths, derive its area polynomial and explain how the multiplication steps match regions in an area model.
Evaluation and interpretation: Evaluate a polynomial model for two inputs, compare the outputs, and explain what the change means in the model's context.
Graph connection: Use a provided polynomial graph and rule to identify a zero, verify it by substitution, and explain why both representations agree.
Create and defend: Write two different polynomials of the same degree that satisfy given constraints, then defend why both meet every condition.

Evidence of Learning

Strong evidence of learning includes accurate use of the words term, coefficient, variable, exponent, constant, degree, and like terms; correct classification of school-level one-variable polynomials; reliable simplification by combining like terms; accurate evaluation by substitution; and correct use of addition, subtraction, and the distributive property.

Useful products include a vocabulary model, a sorting explanation, an area model, an error-analysis poster, a graph-and-rule comparison, or a short tutorial video. Transfer is shown when you can use the same ideas in a new setting, such as translating a geometric situation into a polynomial, checking a graph with substitution, or explaining why an incorrect method fails.

A complete solution should show enough reasoning that another learner can follow the steps, not only the final answer.

OERs on the Topic

The English Wikipedia article below provides a broader reference on polynomials and links to related mathematical ideas.

Linked Learning Areas

Polynomials

Polynomials connect arithmetic, algebra, geometry, functions, graphing, and mathematical modeling. In Grades 7–8, the most important goal is to understand structure: how terms are built, which terms can combine, how operations change an expression, and how a symbolic rule can represent numerical or geometric relationships.

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