Chapter 1 · Algebraic Proofs · Concept 1
1-1 Binomial Theorem
Consider the first rows of Pascal's triangle, and notice how the number of terms changes in each horizontal row. If you follow this pattern, how many terms will there be in the tenth row?
Guiding question
How can algebraic formulas and combinatorial coefficients predict the precise coefficient of any term in an expanded polynomial (x1 + x2)n without performing sequential multiplication?
Learning outcome. Applies the binomial theorem to find the expansion of algebraic expressions and to determine the coefficients of certain terms.
Row n has n + 1 numbers — so the tenth row, which is row n = 9, has 10 terms. Those numbers are exactly the coefficients of (a + b)n.
Calculating nCr
When expanding (a + b)n, the term with an−r br is
nCr an−r br
Read the two exponents as a pair that must always add to n. If one part is used r times, the other is used the remaining n − r times — there is nowhere else for the factors to go.
Expansion of (a + b)n — the Binomial Theorem
(a + b)n = nC0an + nC1an−1b + nC2an−2b2 + …… + nCran−rbr + …… + nCnbn
This one box does every example on these pages, and every question in the Exercise. Change a number and the whole expansion changes with it. Then tap any single term. It shows you the four lines you would write on paper for that term.
Leave the second letter empty to make that part a plain number — that is what turns (2x² − 3)⁷ into a one-letter question. Fractions like 1/3 are allowed and stay exact.
Tap any term to see where it came from.
| r | nCr | Coefficient | Term |
|---|---|---|---|
| 0 | 1 | 128 | 128x14 |
| 1 | 7 | -1344 | -1344x12 |
| 2 | 21 | 6048 | 6048x10 |
| 3 | 35 | -15120 | -15120x8 |
| 4 | 35 | 22680 | 22680x6 |
| 5 | 21 | -20412 | -20412x4 |
| 6 | 7 | 10206 | 10206x2 |
| 7 | 1 | -2187 | -2187 |
These are the book's own two questions. Do not read the steps yet. Try each one first, then open it and check whether you did the same thing.
1. Find the coefficient of x4 in the expansion of (2x2 − 3)7
- 1
Name the two parts.
first part = 2x2 · second part = −3The minus sign belongs to the number. It is −3, not 3.
Set the box above to (2x² − 3)⁷ and tap the term at r = 5. You will see these same lines.
2. Find the expansion of (3x − 2)4
- 1
Name the two parts again.
first part = 3x · second part = −2Same first move every time. Always take the minus with the number.
A mirror. The bar at r is the same height as the bar at n − r. That is nCr = nCn−r, and it is why you may count from whichever end is easier.
A peak. The largest coefficient sits in the middle — and the row adds to 28 = 256. You will prove that in lesson 1-3.
Work with a partner. Solve the Try section with your group, then talk about the answer. You can send any question below to your notebook. Tap it, and it is waiting there for you.
1. Find the coefficient of the specified term in [ ].
2. Find the expansion of the following expressions.
1. Find the coefficient of the specified term in [ ].
2. Find the expansion of the following expressions.
If (1 + x)5 = a + bx + cx2 + dx3 + ex4 + fx5, then find the numerical value of a + b + c + d + e + f.
The minus sign is part of the number.
In (x − 2y)⁶ the second part is −2y, not 2y. Bracket it: (−2y)² = +4y², but (−2y)³ = −8y³. Forgetting this makes every second term the wrong sign.
The coefficient is not just nCr.
For (2x² − 3)⁷ the answer is ₇C₅ × 2² × (−3)⁵, not ₇C₅ on its own. Both parts bring their own numbers with them.
Match the exponent, not the position.
To get x⁶ from (3a² − 2)⁸ you need a² three times — so n − r = 3 and r = 5. Counting terms instead of exponents is the commonest slip on this page.
n + 1 terms, not n.
Row n of Pascal's triangle has n + 1 entries, because r runs from 0 to n. The tenth row of the figure is n = 9.
You will need this at the start of lesson 1-2.
Keep reading — this lesson marks itself done automatically.