If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Operations with polynomials | Lesson

A guide to operations with polynomials on the digital SAT

What are polynomial expressions?

A polynomial expression has one or more terms with a coefficient, a variable base, and an exponent.
In the expression 3x^4, 3 is the coefficient, x is the base, and 4 is the exponent.
3x4 is a
. We'll also frequently see
and
.
  • 3x4+2x is a binomial. The exponent of the term 2x is 1 (x=x1).
  • 3x4+2x+7 is a trinomial. 7 is a constant term. We can also think of 7 as an exponential term with an exponent of 0. Since x0=1, 7 is equivalent to 7x0.
In this lesson, we'll learn to add, subtract, and multiply polynomials.
You can learn anything. Let's do this!

How do I add and subtract polynomials?

Adding polynomials

Khan Academy video wrapper
Adding polynomialsSee video transcript

What should I be careful of when adding and subtracting polynomials?

While we can add and subtract any polynomials, we can only combine like terms, which must have:
  • The same variable base
  • The same exponent
For example, we can combine the terms 2x3 and 4x3 because they have the same variable base, x, and the same exponent, 3. However, we cannot combine the terms 2x2 and 2x3 because they have different exponents, 2 and 3.
When we combine like terms, only the coefficients change. Both the base and the exponent remain the same. For example, when adding 2x3 and 4x3, the x3 part of the terms remain the same, and we add only 2 and 4 when combining the terms:
2x3+4x3=(2+4)x3=6x3
When subtracting polynomials, make sure to distribute the negative sign as needed. For example, when subtracting the polynomial 2x27, the negative sign from the subtraction is distributed to both 2x2 and 7, which means:
5x2(2x27)=5x2+(1)(2x2)+(1)(7)=5x2+2x2+7=(5+2)x2+7=7x2+7
Subtracting 2x27 is equivalent to adding 2x2+7!
To add or subtract two polynomials:
  1. Group like terms.
  2. For each group of like terms, add or subtract the coefficients while keeping both the base and the exponent the same.
  3. Write the combined terms in order of decreasing power.

Try it!

TRY: Match the equivalent expressions
Because 9x2 and 3x2 have
and
, the two terms
into a single term.
Because 4y4 and 4y have
, the two terms
into a single term.


TRY: Match the equivalent expressions
Match each polynomial expression below with an equivalent expression.
1


How do I multiply polynomials?

Multiplying binomials

Khan Academy video wrapper
Multiplying binomialsSee video transcript

What should I be careful of when multiplying polynomials?

When multiplying two polynomials, we must make sure to distribute each term of one polynomial to all the terms of the other polynomial. For example:
(ax+b)(cx+d)=(ax)(cx+d)+(b)(cx+d)=(ax)(cx)+(ax)(d)+(b)(cx)+(b)(d)
The total number of products we need to calculate is equal to the product of the number of terms in each polynomial. Multiplying two binomials requires 22=4 products, as shown above. Multiplying a monomial and a trinomial requires 13=3 products; multiplying a binomial and a trinomial requires 23=6 products.
When multiplying two binomials, we can also use the mnemonic FOIL to account for all four multiplications. For (ax+b)(cx+d):
  1. Multiply the First terms (axcx)
  2. Multiply the Outer terms (axd)
  3. Multiply the Inner terms (bcx)
  4. Multiply the Last terms (bd)
When multiplying terms of polynomial expressions with the same base:
  1. Multiply the coefficients, or multiply the coefficient and the constant.
  2. Keep the base the same.
  3. Add the exponents.
axmbxn=abxm+nabxn=abxn
To multiply two polynomials:
  1. Distribute the terms.
  2. Multiply the distributed terms according to the exponent rules above.
  3. Group like terms.
  4. For each group of like terms, add or subtract the coefficients while keeping both the base and the exponent the same.
  5. Write the combined terms in order of decreasing power.

Let's look at some examples!

What is the product of 2x1 and x5 ?

What is the product of 3x and x24x+9 ?

Try it!

TRY: multiply two terms
When multiplying 3x and 2x2, we
the coefficients of the terms and
the exponents of x.
(3x)(2x2)=


TRY: Multiply two binomials using FOIL
Use the table below to FOIL (8x3)(x2+1).
TermExpressionProduct
First8xx2
Outer
8x
Inner3x2
Last
3


Your turn!

Practice: add two binomials
Which of the following is the sum of x2+5 and 2x2+4x ?
Choose 1 answer:


Practice: subtract two polynomials to find a coefficient
(9x2+5x1)(6x24x)=ax2+bx+c
The equation above is true for all x, where a, b and c are constants. What is the value of b ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi


Practice: multiply two binomials
Which of the following is equivalent to (x+3)(2x5) ?
Choose 1 answer:


Practice: multiply two binomials with symbolic coefficients
(ax+3)(bx+2)=9x2+21x+6
In the equation above, a and b are constants. What is the value of ab ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi


Things to remember

The mnemonic FOIL for multiplying two binomials:
  1. Multiply the First terms
  2. Multiply the Outer terms
  3. Multiply the Inner terms
  4. Multiply the Last terms
axmbxn=abxm+nabxn=abxn

Want to join the conversation?