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The Binomial Theorem

Author: Sophia

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In this lesson, you will learn about Pascal’s triangle and the binomial theorem, which is used to expand binomials raised to integer powers. Specifically, this lesson will cover:

1. Pascal’s Triangle and Binomial Coefficients
2. Using the Binomial Theorem to Expand Powers of Binomials

1. Pascal’s Triangle and Binomial Coefficients

Recall that a binomial is a polynomial with two terms. While we have raised binomials to powers 2 and sometimes 3, raising a binomial to a higher power would be very tedious and time-consuming. We will explore a method to expand open parentheses x plus y close parentheses to the power of n without performing multiplication. Such an expansion is called a binomial expansion.

The table below shows the expansions of open parentheses x plus y close parentheses to the power of n comma assuming x plus y not equal to 0. If you like, you can verify these for yourself by performing the necessary multiplications.

and Its Expansion	Coefficients in Order of Descending Powers of x
	1
	1, 1
	1, 2, 1
	1, 3, 3, 1
	1, 4, 6, 4, 1

Notice that the coefficient column makes up a triangular pattern. This pattern is called Pascal’s triangle, and the numbers in the triangle are called binomial coefficients since they are coefficients of terms in the expansions of open parentheses x plus y close parentheses to the power of n comma where n is a nonnegative integer.

A more formal version of Pascal’s triangle is shown below.

A Pascal’s triangle with eight rows of numbers, representing the binomial coefficients. The binomial coefficients in each row are as follows: First row: 1; Second row: 1 and 1; Third row: 1, 2, and 1; Fourth row: 1, 3, 3, and 1; Fifth row: 1, 4, 6, 4, and 1; Sixth row: 1, 5, 10, 10, 5, and 1; Seventh row: 1, 6, 15, 20, 15, 6, and 1; Eighth row: 1, 7, 21, 35, 35, 21, 7, 1

Properties of Pascal’s triangle:

Each row starts and ends with a “1”.
If we consider the top row as row zero, the coefficients of are in row n. For example, row 3 has coefficients 1, 3, 3, 1 and
The second (and second to last) number in each row corresponds to the power of
After the first row, every number between the 1’s lies below two numbers in the previous row, and is the sum of those numbers. This property helps to build the triangle.

try it

Consider Pascal’s triangle as pictured above.

What is the next row of Pascal’s triangle?

1, 8, 28, 56, 70, 56, 28, 8, 1

EXAMPLE

Use Pascal’s triangle to write the expansion of open parentheses x plus y close parentheses to the power of 5 colon

From the triangle, the coefficients of the terms are 1, 5, 10, 10, 5, 1.
The first term of the expansion is
In subsequent terms, the powers of x decrease by 1, while the power of y increases by 1.
Then, the variable terms are and

Now, incorporating the coefficients, the expansion of open parentheses x plus y close parentheses to the power of 5

x to the power of 5 plus 5 x to the power of 4 y plus 10 x cubed y squared plus 10 x squared y cubed plus 5 x y to the power of 4 plus y to the power of 5.

try it

Consider Pascal’s triangle as pictured above.

Write the expanded form of (x + y )⁶.

open parentheses x plus y close parentheses to the power of 6 equals x to the power of 6 plus 6 x to the power of 5 y plus 15 x to the power of 4 y squared plus 20 x cubed y cubed plus 15 x squared y to the power of 4 plus 6 x y to the power of 5 plus y to the power of 6

As it turns out, each binomial coefficient can be calculated by using a formula.

The notation open parentheses table row n row k end table close parentheses is used to represent a binomial coefficient. Relating to Pascal’s triangle, the value of is in row n, position k, where the first position in each row corresponds to k equals 0.

Written in terms of binomial coefficients, the first few rows of Pascal’s triangle is shown below:

$A representation of binomial coefficients using the notation \binom{n}{k}, which is a pair of parentheses with two numbers stacked; n on top and k on the bottom.$

This all said, we have the following formula to compute a binomial coefficient.

formula to know

Binomial Coefficient

$open parentheses table row n row k end table close parentheses equals fraction numerator n factorial over denominator k factorial open parentheses n minus k close parentheses factorial end fraction$

The expression open parentheses table row n row k end table close parentheses is called a combination, and when referring to it, we say “n choose k.”

hint

When computing binomial coefficients, remember that 1 factorial equals 1

and

EXAMPLE

Compute

open parentheses table row 5 row 2 end table close parentheses.

$open parentheses table row 5 row 2 end table close parentheses equals fraction numerator 5 factorial over denominator 2 factorial open parentheses 5 minus 2 close parentheses factorial end fraction$	This is the formula for computing a binomial coefficient.
$equals fraction numerator 5 factorial over denominator 2 factorial 3 factorial end fraction$	Simplify in the denominator.
$equals fraction numerator 5 open parentheses 4 close parentheses open parentheses 3 close parentheses open parentheses 2 close parentheses open parentheses 1 close parentheses over denominator open parentheses 2 close parentheses open parentheses 1 close parentheses open parentheses 3 close parentheses open parentheses 2 close parentheses open parentheses 1 close parentheses end fraction$	Expand each factorial.
$equals fraction numerator 5 open parentheses 4 close parentheses over denominator 2 end fraction$	Cancel each common factor and factor of 1.
	Simplify.

Thus,

open parentheses table row 5 row 2 end table close parentheses equals 10.

To verify this, look at Pascal’s triangle at the beginning of this lesson, go to row 5, position 2 (remember that the first “1” is in the zero position), and that number is 10.

hint

Consider the expression open parentheses table row 7 row 4 end table close parentheses comma

which is equal to $fraction numerator 7 factorial over denominator 4 factorial 3 factorial end fraction.$ Notice that the “4” and “3” in the denominator add up to 7, the number whose factorial is in the numerator. This is a nice “shortcut” to computing binomial coefficients since you can quickly figure out the other term in the denominator rather than substituting n and k directly into the formula.

try it

Consider the binomial coefficient

Find the value of the binomial coefficient.

$open parentheses table row 8 row 3 end table close parentheses equals fraction numerator 8 factorial over denominator 3 factorial 5 factorial end fraction equals 56$

did you know

A binomial coefficient is also used to solve problems in counting and probability. The expression open parentheses table row n row k end table close parentheses

open parentheses table row n row k end table close parentheses

gives the number of distinct groups of k objects that can be selected from a group of n objects, where order does not matter. For example, the expression open parentheses table row 10 row 3 end table close parentheses

open parentheses table row 10 row 3 end table close parentheses

gives the number of groups of three students that can be selected from a group of 10 students, where no group contains the same three students and the order of the selection of the students does not matter.

terms to know

Binomial Expansion

The expanded form of a binomial that is raised to a nonnegative integer power.

Binomial Coefficient

The coefficients of the terms in the binomial expansion of open parentheses x plus y close parentheses to the power of n.

2. Using the Binomial Theorem to Expand Powers of Binomials

The binomial theorem can be used to find the expansion of a binomial that is raised to a nonnegative integer power.

formula to know

Binomial Theorem

For nonnegative integer n, open parentheses x plus y close parentheses to the power of n equals sum from k equals 0 to n of open parentheses table row n row k end table close parentheses x to the power of n minus k end exponent y to the power of k.

open parentheses x plus y close parentheses to the power of n equals sum from k equals 0 to n of open parentheses table row n row k end table close parentheses x to the power of n minus k end exponent y to the power of k.

Notice that the “k” in the binomial coefficient corresponds to the power of the second term of the binomial.

Let’s first use the theorem to verify a previous result.

EXAMPLE

Use the binomial theorem to find the expansion of open parentheses x plus y close parentheses to the power of 4.

First, substitute n equals 4 colon

open parentheses x plus y close parentheses to the power of 4 equals sum from k equals 0 to 4 of open parentheses table row 4 row k end table close parentheses x to the power of 4 minus k end exponent y to the power of k

Next, expand the summation:

equals open parentheses table row 4 row 0 end table close parentheses x to the power of 4 minus 0 end exponent y to the power of 0 plus open parentheses table row 4 row 1 end table close parentheses x to the power of 4 minus 1 end exponent y to the power of 1 plus open parentheses table row 4 row 2 end table close parentheses x to the power of 4 minus 2 end exponent y squared plus open parentheses table row 4 row 3 end table close parentheses x to the power of 4 minus 3 end exponent y cubed plus open parentheses table row 4 row 4 end table close parentheses x to the power of 4 minus 4 end exponent y to the power of 4

Then, remembering that x to the power of 1 equals x

and

simplify each exponent and evaluate each binomial coefficient:

equals open parentheses table row 4 row 0 end table close parentheses x to the power of 4 plus open parentheses table row 4 row 1 end table close parentheses x cubed y plus open parentheses table row 4 row 2 end table close parentheses x squared y squared plus open parentheses table row 4 row 3 end table close parentheses x y cubed plus open parentheses table row 4 row 4 end table close parentheses y to the power of 4

equals open parentheses 1 close parentheses x to the power of 4 plus open parentheses 4 close parentheses x cubed y plus open parentheses 6 close parentheses x squared y squared plus open parentheses 4 close parentheses x y cubed plus open parentheses 1 close parentheses y to the power of 4

equals x to the power of 4 plus 4 x cubed y plus 6 x squared y squared plus 4 x y cubed plus y to the power of 4

Thus,

open parentheses x plus y close parentheses to the power of 4 equals x to the power of 4 plus 4 x cubed y plus 6 x squared y squared plus 4 x y cubed plus y to the power of 4 comma

which coincides with the result shown earlier.

We can use the binomial theorem to expand more complicated binomials that are raised to a power.

watch

This video shows how to use the binomial theorem to write the expansion of open parentheses 2 x minus 3 y close parentheses to the power of 4.

try it

Consider the expression open parentheses 4 x plus y close parentheses to the power of 5.

Use the binomial theorem to write the expansion.

According to the binomial theorem:

open parentheses 4 x plus y close parentheses to the power of 5 equals open parentheses table row 5 row 0 end table close parentheses open parentheses 4 x close parentheses to the power of 5 y to the power of 0 plus open parentheses table row 5 row 1 end table close parentheses open parentheses 4 x close parentheses to the power of 4 y to the power of 1 plus open parentheses table row 5 row 2 end table close parentheses open parentheses 4 x close parentheses cubed y squared plus open parentheses table row 5 row 3 end table close parentheses open parentheses 4 x close parentheses squared y cubed plus open parentheses table row 5 row 4 end table close parentheses open parentheses 4 x close parentheses to the power of 1 y to the power of 4 plus open parentheses table row 5 row 5 end table close parentheses open parentheses 4 x close parentheses to the power of 0 y to the power of 5

By either evaluating the binomial coefficients or using Pascal’s triangle, the values of the binomial coefficients are (in order) 1, 5, 10, 10, 5, and 1.

Then, by also evaluating the powers of x, the expression simplifies to:

open parentheses 4 x plus y close parentheses to the power of 5 equals open parentheses 1 close parentheses 4 to the power of 5 x to the power of 5 open parentheses 1 close parentheses plus 5 open parentheses 4 close parentheses to the power of 4 x to the power of 4 y plus 10 open parentheses 4 close parentheses cubed x cubed y squared plus 10 open parentheses 4 close parentheses squared x squared y cubed plus 5 open parentheses 4 close parentheses x y to the power of 4 plus 1 open parentheses 1 close parentheses y to the power of 5
space space space space space space space space space space space space space space space space equals 1024 x to the power of 5 plus 1280 x to the power of 4 y plus 640 x cubed y squared plus 160 x squared y cubed plus 20 x y to the power of 4 plus y to the power of 5

The binomial theorem can also be used to find the coefficients of specific terms in a binomial expansion.

EXAMPLE

Find the coefficient of x to the power of 4 y to the power of 5

in the binomial expansion of open parentheses 5 x plus 2 y close parentheses to the power of 9.

	Start with the binomial expansion of since we want to expand another binomial to the 9th power.
	Replace with and b with to obtain the binomial expansion of

Consider now the terms of the expansion, which all have the form open parentheses table row 9 row k end table close parentheses open parentheses 5 x close parentheses to the power of 9 minus k end exponent open parentheses 2 y close parentheses to the power of k.

open parentheses table row 9 row k end table close parentheses open parentheses 5 x close parentheses to the power of 9 minus k end exponent open parentheses 2 y close parentheses to the power of k.

To produce a term with x to the power of 4 y to the power of 5

as the variable part, let k equals 5.

	Substitute into Remember that the power of y determines the value of k. Since the term contains this means
	Expand the exponential factors.
	Evaluate the binomial coefficient, then group the factors with all constants together and all variables together.
	Simplify.

Therefore, the coefficient of x to the power of 4 y to the power of 5

in the expansion of open parentheses 5 x plus 2 y close parentheses to the power of 9

is 2,520,000.

try it

Consider the expression open parentheses 3 x minus 4 y close parentheses to the power of 7.

Determine the coefficient of x ⁵y ² in the binomial expansion.

The coefficient of x to the power of 5 y squared

open parentheses table row 7 row 2 end table close parentheses open parentheses 3 close parentheses to the power of 5 open parentheses short dash 4 close parentheses squared.

Evaluating the binomial coefficient, $open parentheses table row 7 row 2 end table close parentheses equals fraction numerator 7 factorial over denominator 2 factorial 5 factorial end fraction equals 21.$

Then,

summary

In this lesson, you learned that the binomial theorem is used to find binomial expansions of open parentheses x plus y close parentheses to the power of n

without performing multiplications on binomials. Each term in a binomial expansion has a numerical factor called a binomial coefficient. You also learned that the binomial coefficients are visually represented using Pascal’s triangle and can be found by evaluating open parentheses table row n row k end table close parentheses.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Terms to Know

Binomial Coefficient: The coefficients of the terms in the binomial expansion of $open parentheses x plus y close parentheses to the power of n.$
Binomial Expansion: The expanded form of a binomial that is raised to a nonnegative integer power.

Formulas to Know

Binomial Coefficient: $open parentheses table row n row k end table close parentheses equals fraction numerator n factorial over denominator k factorial open parentheses n minus k close parentheses factorial end fraction$
Binomial Theorem: For nonnegative integer n, $open parentheses x plus y close parentheses to the power of n equals sum from k equals 0 to n of open parentheses table row n row k end table close parentheses x to the power of n minus k end exponent y to the power of k.$

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The Binomial Theorem

Table of Contents

1. Pascal’s Triangle and Binomial Coefficients

2. Using the Binomial Theorem to Expand Powers of Binomials