Expand binomials using the binomial theorem

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Number and Algebra

Using coefficients, powers, and structure to expand expressions efficiently

A binomial is an expression with two terms, such as

x+3

or

2x-1

. Multiplying a binomial by itself works for small powers, but repeated multiplication becomes lengthy. The binomial theorem gives a pattern for the coefficients and powers in the expansion. This lesson reviews the needed algebra, develops that pattern, and applies it to exact expansions and checks.

What you will learn

Recognize a binomial and identify its terms and exponent.
Use the binomial theorem to expand a binomial raised to a non-negative integer power.
Find a particular term or coefficient without expanding every term.
Check an expansion using substitution or graphing technology.

1. Prior knowledge: powers and coefficients

An exponent tells how many equal factors are multiplied:

(x+2)^3=(x+2)(x+2)(x+2)

. Expanding means multiplying out the brackets and collecting like terms. For example,

(x+2)^2=x^2+4x+4

.

The binomial theorem extends this process to any non-negative integer exponent

n

. The number of terms in the expansion is

n+1

. The powers of the first term decrease from

n

to

0

, while the powers of the second term increase from

0

to

n

.

The symbol

{n \choose r}

, read as “

n

choose

r

,” is a coefficient. It counts the ways to choose

r

items from

n

items, and can be calculated using factorials. Here,

n

and

r

are whole numbers with

0\le r\le n

;

0!=1

.

{n \choose r}=\frac{n!}{r!(n-r)!}

A negative sign is part of a term, so it must be carried through its powers.
In each term, the exponents of the two binomial terms add to $n$ .

2. The theorem and its pattern

For a binomial

(a+b)^n

, the general term uses the coefficient

{n\choose r}

, the power

a^{n-r}

, and the power

b^r

. Add these terms as

r

runs from

0

to

n

. This is the binomial theorem for a non-negative integer exponent.

For instance, the coefficients for power

4

are 1,4,6,4,1. They are the values

{4\choose 0},{4\choose 1},\ldots,{4\choose 4}

. The corresponding powers move from

a^4

to

a^0

and from

b^0

to

b^4

.

If the second term is negative, substitute that negative term for

b

. Its successive powers alternate in sign: odd powers are negative and even powers are positive. Write terms in descending powers of the first term, and combine like terms if the binomial terms contain the same variable.

(a+b)^n=\sum_{r=0}^{n}{n\choose r}a^{n-r}b^r

There are $n+1$ terms before any like terms are collected.
The coefficient and both powers must be included in each term.

3. Reading an expansion in different ways

Algebraically, the theorem produces an exact polynomial. Numerically, substituting a value for the variable gives a value for the original power and for its expanded form; the two results must agree. This provides a useful check, but one substitution cannot prove that two expressions are identical.

Graphically, the original expression and the expanded polynomial should have the same graph over the real-number domain. A graphing calculator can plot both expressions together or compare their values at several inputs. This is a check for errors, not a replacement for showing the expansion.

In a simple context, if a square has side length

x+2

units, its area is

(x+2)^2

square units. Expanding gives an equivalent expression for that area. The context may restrict

x

so side lengths are non-negative; the algebraic identity itself holds for every real

x

.

(x+2)^2=x^2+4x+4

Use algebra to establish the expansion, then use numerical or graphical checks to detect mistakes.
A context can impose restrictions on a variable even when the algebraic identity is valid more broadly.

4. A reliable expansion routine

Identify the first term, second term, and exponent before writing any terms. Use the coefficient row for that exponent, or calculate each coefficient with

{n\choose r}

. Then write each term with decreasing powers of the first term and increasing powers of the second.

Check that there are

n+1

terms, that each term’s exponents add to

n

, and that signs are consistent. If the question asks for one coefficient only, identify which value of

r

gives the required power; a complete expansion may not be necessary.

A calculator check is most useful after the symbolic work: enter the original power and the proposed polynomial, then compare their values at several real inputs. Keep the written method visible, since a calculator comparison alone does not explain why the expansion has those coefficients.

{n\choose r}a^{n-r}b^r

Organize terms by powers before collecting like terms.
For a requested coefficient, match the variable power to the general-term pattern.

Worked example

Expand a binomial with a positive second term

Expand

(2x+3)^4

.

Identify the terms
Here the first term is $2x$ , the second term is $3$ , and the exponent is $4$ . Use the coefficient row 1,4,6,4,1.
Apply the powers
Decrease the power of $2x$ from $4$ to $0$ and increase the power of $3$ from $0$ to $4$ . The exponents in each product add to $4$ .
$(2x)^4+4(2x)^3(3)+6(2x)^2(3)^2+4(2x)(3)^3+(3)^4$
Simplify each term
Evaluate the numerical factors and write the result in descending powers of $x$ .
$16x^4+96x^3+216x^2+216x+81$

Answer:

(2x+3)^4=16x^4+96x^3+216x^2+216x+81

.

Check: At

x=1

, the original expression is

5^4=625

. The expanded expression gives

16+96+216+216+81=625

.

Worked example

Expand carefully when a term is negative

Expand

(x-2)^5

.

Set the second term
Use the second term as $-2$ , not $2$ . The coefficients for power $5$ are 1,5,10,10,5,1.
Write the terms
The negative second term is raised to successive powers, so its signs alternate. Keep the powers of $x$ in descending order.
$x^5+5x^4(-2)+10x^3(-2)^2+10x^2(-2)^3+5x(-2)^4+(-2)^5$
Simplify
Evaluate each power of $-2$ and collect the resulting terms.
$x^5-10x^4+40x^3-80x^2+80x-32$

Answer:

(x-2)^5=x^5-10x^4+40x^3-80x^2+80x-32

.

Check: At

x=2

, both the original expression and the expansion equal

0

, as expected.

Worked example

Find a particular coefficient

Find the coefficient of

x^3

in

(2x-1)^6

without expanding every term.

Match the power
A general term is ${6\choose r}(2x)^{6-r}(-1)^r$ . Its power of $x$ is $6-r$ . Set that power equal to $3$ .
$6-r=3$
Determine the term
The matching value is $r=3$ . Substitute it into the general term to obtain the entire term containing $x^3$ .
${6\choose 3}(2x)^3(-1)^3$
Read off the coefficient
Since ${6\choose 3}=20$ , the term is $-160x^3$ . The coefficient is the number multiplying $x^3$ .
$20\cdot 8\cdot(-1)=-160$

Answer: The coefficient of

x^3

is

-160

.

Check: The negative sign is correct because the matched term uses the odd power

(-1)^3

.

Common mistakes and how to avoid them

Writing the powers of both terms in descending order.

Correction: The first term’s powers decrease, while the second term’s powers increase. Their exponents add to the original exponent in every term.

Treating

(x-2)^5

as though the second term were positive.

Correction: Use

-2

as the second term. Its powers produce alternating signs.

Stopping after writing the coefficient row.

Correction: A coefficient row gives only the numerical coefficients. Include the powers of both binomial terms as well.

Assuming a matching calculator value proves the expansion.

Correction: Show the theorem-based expansion. Substitution and graphing are checks that can help find errors, not substitutes for the algebra.

Lesson summary

For a non-negative integer exponent $n$ , the binomial expansion has $n+1$ terms.
The term indexed by $r$ is ${n\choose r}a^{n-r}b^r$ .
Track negative terms through their powers, and simplify only after the structure is correct.
To find one coefficient, match the required variable power to $n-r$ and evaluate that term.
Use substitutions or a graphing calculator to check an expansion after completing the algebra.

Check your understanding

Question 1

What is the coefficient of

x^2

in

(x+3)^4

?

18
54
36
81

Show answer and explanation

54

The

x^2

term uses

r=2

:

{4\choose 2}x^2 3^2=6\cdot9x^2=54x^2

.

Question 2

How many terms are in the expansion of

(2x-1)^7

before collecting like terms?

7
8
14
49

Show answer and explanation

8

An exponent of

7

gives

7+1=8

terms.

Question 3

What is the constant term in

(x-2)^4

?

$-16$
$16$
$-8$
$1$

Show answer and explanation

16

The constant term occurs when the power of

x

is zero, giving

(-2)^4=16

.

Key terms

Binomial: An algebraic expression with two terms, such as $x+3$ .
Coefficient: A numerical factor multiplying a term, such as $5$ in $5x^2$ .
Factorial: For a positive whole number $n$ , n! is the product $n(n-1)(n-2)\cdots1$ , and $0!=1$ .
Binomial coefficient: The number ${n\choose r}$ , calculated by $\frac{n!}{r!(n-r)!}$ , used as a coefficient in the expansion.

Continue through IB AA SL

View the complete IB AA SL International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL curriculum and lessons

Tutor support for SL 1.7 · Expand binomials using the binomial theorem

Working on SL 1.7 · Expand binomials using the binomial theorem in IB AA SL? Bring the question and your own attempt. In a live session, you can show each step of your attempt and compare it with a worked example. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث SL 1.7 · بسط دوجمله‌ای‌ها با قضیه دوجمله‌ای در درس IB AA SL، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید گام‌های تلاش خود را نشان دهید و آن را با یک نمونهٔ حل‌شده مقایسه کنید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 1.7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question