Precalculus The Binomial Theorem Test Solutions and Scoring (PDF)

Precalculus math test solutions: The Binomial Theorem – Maths PDF

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

Worked solutions and scoring notes for the 9-question binomial theorem test. Use the scoring line under each solution to award partial credit.

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

Binomial theorem test answers: worked solutions

Question 1 – Evaluate [2 pts]

\(C(9,4)=\frac{9\cdot8\cdot7\cdot6}{4\cdot3\cdot2\cdot1}=126\). \(\frac{7!}{4!3!}=C(7,3)=35\).

\(126\) and \(35\)

Scoring: 1 point each.

Question 2 – Expand [2 pts]

The coefficients are row 5: \(1,5,10,10,5,1\), and all powers of 1 are 1.

\(x^{5}+5x^{4}+10x^{3}+10x^{2}+5x+1\)

Scoring: 1 point for the coefficients, 1 point for the powers.

Question 3 – Expand with a coefficient [2 pts]

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

\(8x^{3}-36x^{2}+54x-27\)

Scoring: 1 point for the setup, 1 point for the simplified result.

Question 4 – A single term [2 pts]

\(k=2\): \(C(6,2)x^{4}\cdot4^{2}=15\cdot16x^{4}\).

\(240x^{4}\)

Scoring: 1 point for k and the setup, 1 point for the term.

Question 5 – Pascal's triangle [3 pts]

Row 4 is 1, 4, 6, 4, 1.

(a) \(1,4,6,4,1\). (b) \(x^{4}+4x^{3}(2)+6x^{2}(4)+4x(8)+16\). (c) Set \(x=1\): \(3^{4}=81\) (check: \(1+8+24+32+16=81\)).

\(1,4,6,4,1\); \(x^{4}+8x^{3}+24x^{2}+32x+16\); \(81\)

Scoring: 1 point for each part.

Question 6 – A coefficient [2 pts]

\(5-k=2\), \(k=3\): \(C(5,3)(3x)^{2}(-2)^{3}=10\cdot9\cdot(-8)x^{2}\).

\(-720\)

Scoring: 1 point for k, 1 point for the coefficient.

Question 7 – Constant term [2 pts]

General term: \(C(9,k)(x^{2})^{9-k}\left(\frac1x\right)^{k}=C(9,k)x^{18-3k}\). The exponent is 0 when \(k=6\).

\(C(9,6)=84\)

Scoring: 1 point for the exponent equation, 1 point for the value.

Question 8 – Probability [3 pts]

15 of the 64 outcomes have 2 heads.

(a) \(C(6,2)=15\). (b) \(2^{6}=64\). (c) \(\frac{15}{64}\approx 0.234\).

\(15\); \(64\); \(0.234\)

Scoring: 1 point for each part.

Question 9 – Approximation [2 pts]

\(1+4(-0.02)+6(-0.02)^{2}=1-0.08+0.0024=0.9224\).

\(0.9224\)

Scoring: 1 point for the terms, 1 point for the sum.

Total score

Total

20 points

Key facts behind this binomial theorem test 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 test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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