Cogito
Precalculus · Chapter 4 · Lesson 2
The Binomial Theorem
Expand a power without multiplying it out.
12 problems · about 22 minutes · TEKS P.5.F
What this lesson teaches
The student expands binomial powers and finds a specified term using binomial coefficients.
- Row n of Pascal’s triangle gives the coefficients of (x + y)ⁿ.
- The powers of x fall while the powers of y rise, always totalling n.
- The term with yʳ is nCr · x^(n−r) · yʳ.
Warm Up
Straightforward practice. Get the method working first.
5 problemsHow many terms does (x + y)⁸ have?
Answer 9
Why 9.
Where do the binomial coefficients come from?
Answer Combinations: choosing which factors contribute a y.
Why They are combinations, nCr.
How many terms in (x + y)⁹?
Answer 10
Why n + 1.
What is 6C2?
Answer 15
Why 720 ÷ (2 × 24).
In (x + y)⁴, what is the coefficient of x²y²?
Answer 6
Why The middle of row 4.
Build It Up
The same ideas with more to keep track of.
3 problemsWhat is the sum of row 3 of Pascal’s triangle: 1, 3, 3, 1?
Answer 8
Why It equals 2³.
In (x + y)⁷, the exponents of every term total what number?
Answer 7
Why They always total n.
In (x + y)⁶, what is the coefficient of x⁵y?
Answer 6
Why 6C1.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut row 5 of Pascal’s triangle in order, left to right.
Answer 1. 1 2. 5 3. 10 4. 10 again
Why The row rises to its peak in the middle.
In (x − y)³, what is the sign of the y³ term? Enter 1 for positive, 2 for negative.
Answer 2
Why (−y)³ keeps the minus.
The Row: How many terms does the expansion of (x + y)⁶ have?
Answer 7
Why 7.
The Coefficient: In (x + y)⁵, what is the coefficient of x³y²?
Answer 10
Why 10.