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Polynomials

Identify the degree and leading coefficient of polynomials.

Identifying the Degree and Leading Coefficient of Polynomials

The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as \(384\pi ,\) is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product \({a}_{i}{x}^{i},\) such as \(384\pi w,\) is a term of a polynomial. If a term does not contain a variable, it is called a constant.

A polynomial containing only one term, such as \(5{x}^{4},\) is called a monomial. A polynomial containing two terms, such as \(2x-9,\) is called a binomial. A polynomial containing three terms, such as \(-3{x}^{2}+8x-7,\) is called a trinomial.

We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.

Example

Try it.

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

ⓐ \(3+2{x}^{2}-4{x}^{3}\)
ⓑ \(5{t}^{5}-2{t}^{3}+7t\)
ⓒ \(6p-{p}^{3}-2\)

Solution

ⓐThe highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, \(-4{x}^{3}.\) The leading coefficient is the coefficient of that term, \(-4.\)
ⓑThe highest power of t is \(5,\) so the degree is \(5.\) The leading term is the term containing that degree, \(5{t}^{5}.\) The leading coefficient is the coefficient of that term, \(5.\)
ⓒThe highest power of p is \(3,\) so the degree is \(3.\) The leading term is the term containing that degree, \(-{p}^{3},\) The leading coefficient is the coefficient of that term, \(-1.\)

Condensed — the full section is in OpenStax College Algebra 2e.

Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, \(5{x}^{2}\) and \(-2{x}^{2}\) are like terms, and can be added to get \(3{x}^{2},\) but \(3x\) and \(3{x}^{2}\) are not like terms, and therefore cannot be added.

Example

Try it.

Find the sum.

\((12{x}^{2}+9x-21)+(4{x}^{3}+8{x}^{2}-5x+20)\)

Solution

\[\begin{array}{ll}4{x}^{3}+(12{x}^{2}+8{x}^{2})+(9x-5x)+(-21+20) & \ \text{Combine like terms}. \\ 4{x}^{3}+20{x}^{2}+4x-1 & \ \text{Simplify}.\end{array}\]

Example

Try it.

Find the difference.

\((7{x}^{4}-{x}^{2}+6x+1)-(5{x}^{3}-2{x}^{2}+3x+2)\)

Solution

\[\begin{array}{ll}7{x}^{4}-{x}^{2}+6x+1-5{x}^{3}+2{x}^{2}-3x-2 & \ \text{Distribute negative sign.} \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+6x-3x+1-2 & \ \text{Group like terms.} \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x-1 & \ \text{Combine/simplify.}\end{array}\]

Multiplying Polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the \(2\) in \(2(x+7)\) to obtain the equivalent expression \(2x+14.\) When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

Example

Try it.

Find the product.

\((2x+1)(3{x}^{2}-x+4)\)

Solution

\[\begin{array}{ll}2x(3{x}^{2}-x+4)+1(3{x}^{2}-x+4) & \ \text{Use the distributive property}. \\ (6{x}^{3}-2{x}^{2}+8x)+(3{x}^{2}-x+4) & \ \text{Multiply}. \\ 6{x}^{3}+(-2{x}^{2}+3{x}^{2})+(8x-x)+4 & \ \text{Combine like terms}. \\ 6{x}^{3}+{x}^{2}+7x+4 & \ \text{Simplify}.\end{array}\]

Condensed — the full section is in OpenStax College Algebra 2e.

Performing Operations with Polynomials of Several Variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:

\[\begin{array}{ll}(a+2b)(4a-b-c) & \\ a(4a-b-c)+2b(4a-b-c) & \ \text{Use the distributive property}. \\ 4{a}^{2}-ab-ac+8ab-2{b}^{2}-2bc & \ \text{Multiply}. \\ 4{a}^{2}+(-ab+8ab)-ac-2{b}^{2}-2bc & \ \text{Combine like terms}. \\ 4{a}^{2}+7ab-ac-2bc-2{b}^{2} & \ \text{Simplify}.\end{array}\]

Example

Try it.

Multiply \((x+4)(3x-2y+5).\)

Solution

Follow the same steps that we used to multiply polynomials containing only one variable.

\[\begin{array}{ll}x(3x-2y+5)+4(3x-2y+5) & \ \text{Use the distributive property}. \\ 3{x}^{2}-2xy+5x+12x-8y+20 & \ \text{Multiply}. \\ 3{x}^{2}-2xy+(5x+12x)-8y+20 & \ \text{Combine like terms}. \\ 3{x}^{2}-2xy+17x-8y+20 & \ \text{Simplify}.\end{array}\]

Key Concepts

A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See .
We can add and subtract polynomials by combining like terms. See and .
To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See .
FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See .
Perfect square trinomials and difference of squares are special products. See and .
Follow the same rules to work with polynomials containing several variables. See .

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

For the following polynomials, identify the degree, the leading term, and the leading coefficient.
1. ⓐ \(3+2{x}^{2}-4{x}^{3}\)
2. ⓑ \(5{t}^{5}-2{t}^{3}+7t\)
3. ⓒ \(6p-{p}^{3}-2\)
Asehoy ny valinteny
1. ⓐThe highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, \(-4{x}^{3}.\) The leading coefficient is the coefficient of that term, \(-4.\)
2. ⓑThe highest power of t is \(5,\) so the degree is \(5.\) The leading term is the term containing that degree, \(5{t}^{5}.\) The leading coefficient is the coefficient of that term, \(5.\)
3. ⓒThe highest power of p is \(3,\) so the degree is \(3.\) The leading term is the term containing that degree, \(-{p}^{3},\) The leading coefficient is the coefficient of that term, \(-1.\)
Identify the degree, leading term, and leading coefficient of the polynomial \(4{x}^{2}-{x}^{6}+2x-6.\)

Asehoy ny valinteny

The degree is 6, the leading term is \(-{x}^{6},\) and the leading coefficient is \(-1.\)
Find the sum.
\((12{x}^{2}+9x-21)+(4{x}^{3}+8{x}^{2}-5x+20)\)

Asehoy ny valinteny

\[\begin{array}{ll}4{x}^{3}+(12{x}^{2}+8{x}^{2})+(9x-5x)+(-21+20) & \ \text{Combine like terms}. \\ 4{x}^{3}+20{x}^{2}+4x-1 & \ \text{Simplify}.\end{array}\]
Find the sum.
\((2{x}^{3}+5{x}^{2}-x+1)+(2{x}^{2}-3x-4)\)

Asehoy ny valinteny

\(2{x}^{3}+7{x}^{2}-4x-3\)
Find the difference.
\((7{x}^{4}-{x}^{2}+6x+1)-(5{x}^{3}-2{x}^{2}+3x+2)\)

Asehoy ny valinteny

\[\begin{array}{ll}7{x}^{4}-{x}^{2}+6x+1-5{x}^{3}+2{x}^{2}-3x-2 & \ \text{Distribute negative sign.} \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+6x-3x+1-2 & \ \text{Group like terms.} \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x-1 & \ \text{Combine/simplify.}\end{array}\]
Find the difference.
\((-7{x}^{3}-7{x}^{2}+6x-2)-(4{x}^{3}-6{x}^{2}-x+7)\)

Asehoy ny valinteny

\(-11{x}^{3}-{x}^{2}+7x-9\)
Find the product.
\((2x+1)(3{x}^{2}-x+4)\)

Asehoy ny valinteny

\[\begin{array}{ll}2x(3{x}^{2}-x+4)+1(3{x}^{2}-x+4) & \ \text{Use the distributive property}. \\ (6{x}^{3}-2{x}^{2}+8x)+(3{x}^{2}-x+4) & \ \text{Multiply}. \\ 6{x}^{3}+(-2{x}^{2}+3{x}^{2})+(8x-x)+4 & \ \text{Combine like terms}. \\ 6{x}^{3}+{x}^{2}+7x+4 & \ \text{Simplify}.\end{array}\]
Find the product.
\((3x+2)({x}^{3}-4{x}^{2}+7)\)

Asehoy ny valinteny

\(3{x}^{4}-10{x}^{3}-8{x}^{2}+21x+14\)
Use FOIL to find the product.
\((2x-18)(3x+3)\)

Asehoy ny valinteny

Find the product of the first terms.
Find the product of the outer terms.
Find the product of the inner terms.
Find the product of the last terms.
\(\begin{array}{ll}6{x}^{2}+6x-54x-54 & \ \text{Add the products}. \\ 6{x}^{2}+(6x-54x)-54 & \ \text{Combine like terms}. \\ 6{x}^{2}-48x-54 & \ \text{Simplify}.\end{array}\)
Use FOIL to find the product.
\((x+7)(3x-5)\)

Asehoy ny valinteny

\(3{x}^{2}+16x-35\)
Expand \({(3x-8)}^{2}.\)

Asehoy ny valinteny

Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.
\[{(3x)}^{2}-2(3x)(8)+{(-8)}^{2}\]
Simplify.\[9{x}^{2}-48x+64.\]
Expand \({(4x-1)}^{2}.\)

Asehoy ny valinteny

\(16{x}^{2}-8x+1\)
Multiply \((9x+4)(9x-4).\)

Asehoy ny valinteny

Square the first term to get \({(9x)}^{2}=81{x}^{2}.\) Square the last term to get \({4}^{2}=16.\) Subtract the square of the last term from the square of the first term to find the product of \(81{x}^{2}-16.\)
Multiply \((2x+7)(2x-7).\)

Asehoy ny valinteny

\(4{x}^{2}-49\)
Multiply \((x+4)(3x-2y+5).\)

Asehoy ny valinteny

Follow the same steps that we used to multiply polynomials containing only one variable.
\[\begin{array}{ll}x(3x-2y+5)+4(3x-2y+5) & \ \text{Use the distributive property}. \\ 3{x}^{2}-2xy+5x+12x-8y+20 & \ \text{Multiply}. \\ 3{x}^{2}-2xy+(5x+12x)-8y+20 & \ \text{Combine like terms}. \\ 3{x}^{2}-2xy+17x-8y+20 & \ \text{Simplify}.\end{array}\]
Multiply \((3x-1)(2x+7y-9).\)

Asehoy ny valinteny

\(6{x}^{2}+21xy-29x-7y+9\)
Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.

Asehoy ny valinteny

The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.
Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.
You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?

Asehoy ny valinteny

Use the distributive property, multiply, combine like terms, and simplify.
State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial.
\(7x-2{x}^{2}+13\)

Asehoy ny valinteny

2
\(14{m}^{3}+{m}^{2}-16m+8\)
\(-625{a}^{8}+16{b}^{4}\)

Asehoy ny valinteny

8
\(200p-30{p}^{2}m+40{m}^{3}\)
\({x}^{2}+4x+4\)

Asehoy ny valinteny

2
\(6{y}^{4}-{y}^{5}+3y-4\)
\((12{x}^{2}+3x)-(8{x}^{2}-19)\)

Asehoy ny valinteny

\(4{x}^{2}+3x+19\)
\((4{z}^{3}+8{z}^{2}-z)+(-2{z}^{2}+z+6)\)
\((6{w}^{2}+24w+24)-(3w-26w+3)\)

Asehoy ny valinteny

\(3{w}^{2}+30w+21\)
\((7{a}^{3}+6{a}^{2}-4a-13)+(-3{a}^{3}-4{a}^{2}+6a+17)\)
\((11{b}^{4}-6{b}^{3}+18{b}^{2}-4b+8)-(3{b}^{3}+6{b}^{2}+3b)\)

Asehoy ny valinteny

\(11{b}^{4}-9{b}^{3}+12{b}^{2}-7b+8\)
\((49{p}^{2}-25)+(16{p}^{4}-32{p}^{2}+16)\)
\((4x+2)(6x-4)\)

Asehoy ny valinteny

\(24{x}^{2}-4x-8\)
\((14{c}^{2}+4c)(2{c}^{2}-3c)\)
\((6{b}^{2}-6)(4{b}^{2}-4)\)

Asehoy ny valinteny

\(24{b}^{4}-48{b}^{2}+24\)
\((3d-5)(2d+9)\)
\((9v-11)(11v-9)\)

Asehoy ny valinteny

\(99{v}^{2}-202v+99\)
\((4{t}^{2}+7t)(-3{t}^{2}+4)\)
\((8n-4)({n}^{2}+9)\)

Asehoy ny valinteny

\(8{n}^{3}-4{n}^{2}+72n-36\)
\({(4x+5)}^{2}\)

Symbols used here

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

imaginary unit
i² = −1.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Polynomials

Identify the degree and leading coefficient of polynomials.
Add and subtract polynomials.
Multiply polynomials.
Use FOIL to multiply binomials.
Perform operations with polynomials of several variables.
Distinguish between a fixed and a growth mindset, and how these ideas may help in learning.
Read through the following illustration based on Dr. Dweck’s work.
It’s important to note that we as individuals do not have a strict fixed or growth mindset at all times. We can lean one way or another in certain situations or when working in different disciplines or areas. For example, a person who often plays video games may feel they can learn any new game that is released and be confident in these abilities, but at the same time avoid sports and are fixed on the idea that they will never excel at physical activities.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Andramo ny anao manokana

Parts of this page are adapted from OpenStax College Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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