Cogito
Integrated Math 3 · Chapter 1 · Lesson 3
Polynomial Identities and the Binomial Theorem
Expanding without multiplying it all out.
12 problems · about 21 minutes · A-APR.C.4, A-APR.C.5
What this lesson teaches
The student applies polynomial identities and expands binomial powers.
- Row n of Pascal's triangle gives the coefficients of (a + b)ⁿ.
- The powers of a fall while the powers of b rise, always totalling n.
- The term with bʳ is nCr · a^(n−r) · bʳ.
Warm Up
Straightforward practice. Get the method working first.
5 problemsHow many terms does (a + b)⁸ have?
Answer 9
Why 9.
Where do the binomial coefficients come from?
Answer Combinations: choosing which factors contribute a b.
Why They are nCr.
How many terms in (a + b)⁹?
Answer 10
Why n + 1.
What is 6C2?
Answer 15
Why 720 ÷ (2 × 24).
In (a + b)⁴, coefficient of a²b²?
Answer 6
Why The middle of row 4.
Build It Up
The same ideas with more to keep track of.
3 problemsIn (a + b)², what is the coefficient of ab?
Answer 2
Why The forgotten middle term.
Sum of row 3 of the triangle: 1, 3, 3, 1?
Answer 8
Why It equals 2³.
In (a + b)⁷, the exponents of every term total what?
Answer 7
Why Always n.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each expression with its factorisation.
Answer a squared minus b squared → (a minus b)(a plus b); a cubed minus b cubed → (a minus b)(a squared plus ab plus b squared); a cubed plus b cubed → (a plus b)(a squared minus ab plus b squared)
Why The sign inside the bracket opposes the outer one.
In (a + b)⁶, coefficient of a⁵b?
Answer 6
Why 6C1.
The Terms: How many terms does (a + b)⁶ have?
Answer 7
Why 7.
The Coefficient: In (a + b)⁵, what is the coefficient of a³b²?
Answer 10
Why 10.