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All courses Algebra II · A-APR.5 12 of 55
Apply the Binomial Theorem for expanding (x+y)^n using Pascal's Triangle or combinatorial reasoning.

Expand a binomial power using Pascal's Triangle

Problem
Use Pascal's Triangle to expand \((x~+~y)^4\).
Pascal's Triangle through row 4, with row 4 highlighted as 1, 4, 6, 4, 1 for (x+y)^4. Apply these coefficients to descending powers of x and ascending powers of y; the expansion is withheld. Open full size
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Hint

Use the Pascal's Triangle row for power \(4\): \(1, 4, 6, 4, 1\).

The powers of \(x\) decrease from \(4\) to \(0\), while the powers of \(y\) increase from \(0\) to \(4\).

Solution walkthrough

01

Select the row for exponent four

\[n=4~->~\text{coefficients}~1,4,6,4,1\]

Pascal's Triangle row 4 supplies five coefficients for a fourth-power binomial. The inspected source visual highlights exactly 1, 4, 6, 4, 1.

02

Build the complementary power pattern

\[x^4y^0,~x^3y^1,~x^2y^2,~x^1y^3,~x^0y^4\]

Across the five terms, the exponent of x decreases from 4 to 0 while the exponent of y increases from 0 to 4. Each pair sums to the total exponent 4.

03

Attach each coefficient to its term

\[1x^4+4x^3y+6x^2y^2+4xy^3+1y^4\]

Match the Pascal coefficients in order with the five power products. Powers of zero reduce the first and last products to x⁴ and y⁴.

04

Write and check the full expansion

\[x^4~+~4x^3y~+~6x^2y^2~+~4xy^3~+~y^4\]

The coefficients are symmetric and sum to 16, which agrees with evaluating (1+1)⁴=16. The expansion is x⁴+4x³y+6x²y²+4xy³+y⁴, choice A.

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Another way

  1. Apply the Binomial Theorem sum from k=0 to 4 using coefficient 4Ck and term x⁴⁻kyk.

!

Common mistake

Do not use the coefficients without the complementary exponent pattern. The x power decreases by one while the y power increases by one in every successive term.

x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴
x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Solution walkthrough video