Apply the Binomial Theorem

Lesson 5 of 7: Arithmetic with Polynomials (+)

In this lesson:

• Build Pascal's Triangle and read off binomial coefficients
• State and apply the Binomial Theorem for any positive integer
• Find specific terms of a binomial expansion without full expansion

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Construct Pascal's Triangle and identify for any row
2. Expand using Pascal's Triangle, correctly applying the exponent pattern
3. Calculate binomial coefficients using
4. Find specific terms of a binomial expansion without full expansion
5. Apply the Binomial Theorem to non-standard binomials like

Recall: Binomial Expansions You Already Know

• — coefficients: 1, 2, 1
• — coefficients: 1, 3, 3, 1
• Each term: -exponent + -exponent =

What is the coefficient of in ?

Without Multiplying, Expand

You may know:

What are the coefficients for ? For ?

Is there a pattern that avoids repeated multiplication?

Prior Identities Are Pascal's Triangle Rows

← coefficients: 1, 2, 1

← coefficients: 1, 3, 3, 1

These coefficient rows are rows 2 and 3 of Pascal's Triangle.

Build Pascal's Triangle: Rows 0 Through 5

Each entry = sum of two above it. Row → coefficients for .

Why Adjacent Entries Add Up

: each term picks or from each of factors.

Coefficient of = ways to choose factors to give = .

Adjacent entries add because .

Use Row 5 to Expand

Row 5 coefficients: 1, 5, 10, 10, 5, 1

Exponent pattern: -exponent decreases from 5 to 0; -exponent increases from 0 to 5.

Sum of exponents in each term equals 5.

Check-In: Expand

Write out Row 6 of Pascal's Triangle, then use it to expand .

Reminder: each entry in Row 6 is the sum of two adjacent entries in Row 5.

Verify that every term has exponents summing to 6.

Pascal's Triangle Has Limits: We Need a Formula

Pascal's Triangle is efficient for small .

But for : you would need to build 21 rows.

Better: compute any coefficient directly using

No building required — just compute the entry you need.

The Binomial Theorem: Formal Statement

Written term by term:

Computing

Key: by definition; ensures .

• (matches Row 4)

Shortcut for small :

Full Expansion Using the Formula:

Probe: Connect the Triangle to the Formula

Row 4: 1, 4, 6, 4, 1

• — —

Verify using the formula.

The th Term: No Full Expansion Needed

The general term of :

The th term uses index — not . First term: .

Key rule: "Find the th term" → set in the formula.

Find the 5th Term of

5th term →

Verify: ✓

Find a Term by Its -Exponent

means (since the exponent on equals ).

Probe: Spot the Off-by-One Error

Question: "Find the 4th term of ."

Two students:

• Student A uses → gets
• Student B uses → gets

Who is correct? Why does the other fail?

Non-Standard Binomials: Full Expression in the Formula

In : the roles of and in the theorem are played by and .

Every term:

— raise the full expression, not just .

Worked: Expand

Treat as . Substitute :

Worked: 4th Term of

4th term →

Two steps: index correction () and inner coefficient ().

Check-In: Apply All Three Techniques

1. Identify : the -exponent is 2, so
2. Write the term:
3. Compute:

Verify the sign: , so the term is positive.

Three Common Errors to Avoid

Off-by-one: "4th term" → use , not

Inner coefficient: , not — raise the full expression

Sign pattern: — odd gives negative; even gives positive

Your Turn: Unscaffolded Term Problem

Find the 4th term of :

• (index correction)

Summary: The Binomial Theorem as a Power Tool

• Pascal's Triangle: fast for small — read row
• formula: works for any directly
• Specific terms: th term → index
• Non-standard: substitute full expressions; track signs

Next: These Coefficients Reappear in Probability

counts the ways to choose items from — combinations.

In the binomial distribution:

Flipping a fair coin 10 times: probability of exactly 6 heads?