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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.
start color #7854ab, 3, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript is a
monomial
. We'll also frequently see
binomials
and
trinomials
.
  • start color #7854ab, 3, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript, start color #7854ab, plus, 2, end color #7854ab, start color #ca337c, x, end color #ca337c is a binomial. The exponent of the term start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c is start color #208170, 1, end color #208170 (start color #ca337c, x, end color #ca337c, equals, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 1, end color #208170, end superscript).
  • start color #7854ab, 3, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript, start color #7854ab, plus, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start color #a75a05, plus, 7, end color #a75a05 is a trinomial. start color #a75a05, 7, end color #a75a05 is a constant term. We can also think of start color #a75a05, 7, end color #a75a05 as an exponential term with an exponent of start color #208170, 0, end color #208170. Since start color #ca337c, x, start superscript, start color #208170, 0, end color #208170, end superscript, end color #ca337c, equals, 1, start color #a75a05, 7, end color #a75a05 is equivalent to start color #7854ab, 7, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 0, end color #208170, end superscript.
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 start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript and start color #7854ab, 4, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript because they have the same variable base, start color #ca337c, x, end color #ca337c, and the same exponent, start color #208170, 3, end color #208170. However, we cannot combine the terms start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 2, end color #208170, end superscript and start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript because they have different exponents, start color #208170, 2, end color #208170 and start color #208170, 3, end color #208170.
When we combine like terms, only the coefficients change. Both the base and the exponent remain the same. For example, when adding start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript and start color #7854ab, 4, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript, the start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript part of the terms remain the same, and we add only start color #7854ab, 2, end color #7854ab and start color #7854ab, 4, end color #7854ab when combining the terms:
2x3+4x3=(2+4)x3=6x3\begin{aligned} \purpleD{2}\maroonD{x}^{\tealE{3}} + \purpleD{4}\maroonD{x}^{\tealE{3}} &=\purpleD{(2+4)}\maroonD{x}^{\tealE{3}} \\\\ &=\purpleD{6}\maroonD{x}^{\tealE{3}} \end{aligned}
When subtracting polynomials, make sure to distribute the negative sign as needed. For example, when subtracting the polynomial minus, 2, x, squared, minus, 7, the negative sign from the subtraction is distributed to both minus, 2, x, squared and minus, 7, which means:
5x2(2x27)=5x2+(1)(2x2)+(1)(7)=5x2+2x2+7=(5+2)x2+7=7x2+7\begin{aligned} &5x^2-(-2x^2-7) \\\\ &=5x^2+(-1)(-2x^2)+(-1)(-7) \\\\ &=5x^2+2x^2+7\\\\ &=(5+2)x^2+7 \\\\ &=7x^2+7 \end{aligned}
Subtracting minus, 2, x, squared, minus, 7 is equivalent to adding 2, x, squared, plus, 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 9, x, squared and 3, x, squared have
and
, the two terms
into a single term.
Because 4, y, start superscript, 4, end superscript and 4, y have
, the two terms
into a single term.


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


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)\begin{aligned} &(\purpleD{ax}+\maroonD{b})(\tealE{cx}+\goldE{d}) \\\\ =&\,(\purpleD{ax})(\tealE{cx}+\goldE{d})+(\maroonD{b})(\tealE{cx}+\goldE{d}) \\\\ =&\,(\purpleD{ax})(\tealE{cx})+(\purpleD{ax})(\goldE{d})+(\maroonD{b})(\tealE{cx})+(\maroonD{b})(\goldE{d}) \end{aligned}
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 2, dot, 2, equals, 4 products, as shown above. Multiplying a monomial and a trinomial requires 1, dot, 3, equals, 3 products; multiplying a binomial and a trinomial requires 2, dot, 3, equals, 6 products.
When multiplying two binomials, we can also use the mnemonic FOIL to account for all four multiplications. For left parenthesis, start color #7854ab, a, x, end color #7854ab, plus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, start color #208170, c, x, end color #208170, plus, start color #a75a05, d, end color #a75a05, right parenthesis:
  1. Multiply the First terms (start color #7854ab, a, x, end color #7854ab, dot, start color #208170, c, x, end color #208170)
  2. Multiply the Outer terms (start color #7854ab, a, x, end color #7854ab, dot, start color #a75a05, d, end color #a75a05)
  3. Multiply the Inner terms (start color #ca337c, b, end color #ca337c, dot, start color #208170, c, x, end color #208170)
  4. Multiply the Last terms (start color #ca337c, b, end color #ca337c, dot, start color #a75a05, d, end color #a75a05)
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\begin{aligned} \purpleD{a}\maroonD{x}^\tealE{m} \cdot \goldE{b}\maroonD{x}^\redE{n} &= \purpleD{a}\goldE{b}\cdot \maroonD{x}^{\tealE{m}+\redE{n}} \\\\ \purpleD{a}\cdot \goldE{b}\maroonD{x}^\redE{n}&=\purpleD{a}\goldE{b}\cdot \maroonD{x}^\redE{n} \end{aligned}
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 2, x, minus, 1 and x, minus, 5 ?

What is the product of 3, x and x, squared, minus, 4, x, plus, 9 ?

Try it!

TRY: multiply two terms
When multiplying 3, x and 2, x, squared, we
the coefficients of the terms and
the exponents of x.
left parenthesis, 3, x, right parenthesis, left parenthesis, 2, x, squared, right parenthesis, equals


TRY: Multiply two binomials using FOIL
Use the table below to FOIL left parenthesis, 8, x, minus, 3, right parenthesis, left parenthesis, x, squared, plus, 1, right parenthesis.
TermExpressionProduct
First8, x, dot, x, squared
Outer
8, x
Innerminus, 3, dot, x, squared
Last
minus, 3


Your turn!

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


Practice: subtract two polynomials to find a coefficient
left parenthesis, 9, x, squared, plus, 5, x, minus, 1, right parenthesis, minus, left parenthesis, 6, x, squared, minus, 4, x, right parenthesis, equals, a, x, squared, plus, b, x, plus, 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, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


Practice: multiply two binomials
Which of the following is equivalent to left parenthesis, x, plus, 3, right parenthesis, left parenthesis, 2, x, minus, 5, right parenthesis ?
Choose 1 answer:
Choose 1 answer:


Practice: multiply two binomials with symbolic coefficients
left parenthesis, a, x, plus, 3, right parenthesis, left parenthesis, b, x, plus, 2, right parenthesis, equals, 9, x, squared, plus, 21, x, plus, 6
In the equation above, a and b are constants. What is the value of a, b ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


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\begin{aligned} \purpleD{a}\maroonD{x}^\tealE{m} \cdot \goldE{b}\maroonD{x}^\redE{n} &= \purpleD{a}\goldE{b}\cdot \maroonD{x}^{\tealE{m}+\redE{n}} \\\\ \purpleD{a}\cdot \goldE{b}\maroonD{x}^\redE{n}&=\purpleD{a}\goldE{b}\cdot \maroonD{x}^\redE{n} \end{aligned}

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