Precalculus The Binomial Theorem Worksheet Solutions (PDF)

Precalculus math worksheet solutions: The Binomial Theorem – Maths PDF

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This binomial theorem answers for Precalculus covers the binomial theorem (Common Core A-APR.C.5), with a free PDF to print.

Use this key to check each expansion and coefficient. Solutions show the row of Pascal’s triangle or the general term used, with final answers in bold.

Precalculus · Common Core State Standards: A-APR.C.5

Binomial theorem answers: exercises 1 to 10

Exercise 1 – Factorials

\(6!=720\). \(0!=1\) by definition. \(\frac{8!}{6!}=8\cdot7=56\).

\(720,\ 1,\ 56\)

Exercise 2 – Binomial coefficients

\(C(7,3)=\frac{7\cdot6\cdot5}{3\cdot2\cdot1}=35\). \(C(10,2)=\frac{10\cdot9}{2}=45\). \(C(9,9)=1\). \(C(12,10)=C(12,2)=\frac{12\cdot11}{2}=66\).

\(35,\ 45,\ 1,\ 66\)

Exercise 3 – Pascal's triangle

Row 6 is shaded.

Add neighbors in row 5 (\(1,5,10,10,5,1\)): \(1,\ 1+5,\ 5+10,\ 10+10,\ 10+5,\ 5+1,\ 1\).

\(1,\ 6,\ 15,\ 20,\ 15,\ 6,\ 1\)

Exercise 4 – Expand (x + y)^4

Row 4 of the triangle is \(1,4,6,4,1\).

\(x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}\)

Exercise 5 – Expand (x + 2)^5

C(5,k), powers of x and powers of 2.

Multiply the row-5 coefficients by powers of 2 and add the powers of \(x\).

\(x^{5}+10x^{4}+40x^{3}+80x^{2}+80x+32\)

Exercise 6 – Expand (3x + 1)^4

\((3x)^{4}+4(3x)^{3}(1)+6(3x)^{2}(1)^{2}+4(3x)(1)^{3}+1^{4}\).

\(81x^{4}+108x^{3}+54x^{2}+12x+1\)

Exercise 7 – Expand (x – 3)^3

Row 3 is \(1,3,3,1\). \(x^{3}+3x^{2}(-3)+3x(9)+(-27)\).

\(x^{3}-9x^{2}+27x-27\)

Exercise 8 – Find a single term

The 4th term has \(k=3\): \(C(8,3)\,x^{5}\cdot3^{3}=56\cdot27\,x^{5}\).

\(1512x^{5}\)

Exercise 9 – Find a coefficient

Power \(x^{3}\) needs \(6-k=3\), so \(k=3\): \(C(6,3)(2x)^{3}(1)^{3}=20\cdot8x^{3}\).

160

Exercise 10 – Coefficient with a negative

\(x^{4}\) means \(7-k=4\), \(k=3\). \(C(7,3)(-2)^{3}=35(-8)\).

\(-280\)

Exercises 11 and up

Exercise 11 – Constant term

The general term is \(C(6,k)x^{6-k}\left(\frac{2}{x}\right)^{k}=C(6,k)2^{k}x^{6-2k}\). The exponent is 0 when \(k=3\): \(C(6,3)\cdot8=20\cdot8\).

160

Exercise 12 – Sum of coefficients

Set \(x=1\): \((3-1)^{5}=2^{5}\).

Exercise 13 – Row sums

The bars are 1, 5, 10, 10, 5, 1.

\(1+5+10+10+5+1=32=2^{5}\). This matches \((1+1)^{5}\).

Exercise 14 – Middle term

There are 11 terms; the middle is \(k=5\): \(C(10,5)x^{5}y^{5}\).

\(252x^{5}y^{5}\)

Exercise 15 – Approximate a power

\(1+5(0.02)+10(0.02)^{2}+10(0.02)^{3}=1+0.1+0.004+0.00008=1.10408\).

\(1.10408\), very close to \(1.10408\)

Exercise 16 – Complex number

\(1+4i+6i^{2}+4i^{3}+i^{4}=1+4i-6-4i+1\).

\(-4\)

Exercise 17 – Choose a committee

Order does not matter: \(C(8,3)=\frac{8\cdot7\cdot6}{3\cdot2\cdot1}\).

56 committees

Exercise 18 – Coin tosses

10 of the 32 outcomes have 3 heads.

There are \(2^{5}=32\) equally likely outcomes, and \(C(5,3)=10\) have exactly 3 heads.

\(\frac{10}{32}=\frac{5}{16}\)

Exercise 19 – Free throws

The bar at 4 is about 36 percent.

The term of \((0.7+0.3)^{5}\) with 4 makes is \(C(5,4)(0.7)^{4}(0.3)^{1}=5(0.2401)(0.3)=0.36015\).

\(\approx0.360\)

Exercise 20 – Two variables

Terms: \((2x)^{4}\); \(4(2x)^{3}(-3y)\); \(6(2x)^{2}(-3y)^{2}\); \(4(2x)(-3y)^{3}\); \((-3y)^{4}\).

\(16x^{4}-96x^{3}y+216x^{2}y^{2}-216xy^{3}+81y^{4}\)

Key facts behind this binomial theorem answers

Key facts

n! = n(n – 1)…(1) and 0! = 1; C(n, k) = n!/(k!(n – k)!).
Pascal’s triangle: each entry is the sum of the two above; row n sums to 2^n.
(a + b)^n = sum of C(n, k) a^(n-k) b^k for k = 0 to n.
The general term is T(k+1) = C(n, k) a^(n-k) b^k; signs alternate when b is negative.
Setting x = 1 gives the sum of the coefficients.

How to use this binomial theorem answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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