12.4 Binomial Theorem

Before you get started, take this readiness quiz.

Simplify: $\frac{7 \cdot 6 \cdot 5 \cdot 4}{4 \cdot 3 \cdot 2 \cdot 1} .$
If you missed this problem, review the reference.

Expand: $\left(3 x + 5\right)^{2} .$
If you missed this problem, review the reference.

Expand: $\left(x - y\right)^{2} .$
If you missed this problem, review the reference.

Use Pascal’s Triangle to expand $\left(x + y\right)^{5} .$

Use Pascal’s Triangle to expand $\left(p + q\right)^{7} .$

Use Pascal’s Triangle to expand $\left(x + 2\right)^{4} .$

Use Pascal’s Triangle to expand $\left(x + 1\right)^{6} .$

Use Pascal’s Triangle to expand $\left(2 x - 3\right)^{4} .$

Use Pascal’s Triangle to expand $\left(2 x - 1\right)^{6} .$

Evaluate: ⓐ $\begin{pmatrix} 5 \\ 1 \end{pmatrix}$ ⓑ $\begin{pmatrix} 7 \\ 7 \end{pmatrix}$ ⓒ $\begin{pmatrix} 4 \\ 0 \end{pmatrix}$ ⓓ $\begin{pmatrix} 8 \\ 5 \end{pmatrix} .$

Evaluate each binomial coefficient:

ⓐ $\begin{pmatrix} 6 \\ 1 \end{pmatrix}$ ⓑ $\begin{pmatrix} 8 \\ 8 \end{pmatrix}$ ⓒ $\begin{pmatrix} 5 \\ 0 \end{pmatrix}$ ⓓ $\begin{pmatrix} 7 \\ 3 \end{pmatrix} .$

Evaluate each binomial coefficient:

ⓐ $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ ⓑ $\begin{pmatrix} 11 \\ 11 \end{pmatrix}$ ⓒ $\begin{pmatrix} 9 \\ 0 \end{pmatrix}$ ⓓ $\begin{pmatrix} 6 \\ 5 \end{pmatrix} .$

Use the Binomial Theorem to expand $\left(x + y\right)^{5} .$

Use the Binomial Theorem to expand $\left(m + n\right)^{6} .$

Use the Binomial Theorem to expand $\left(x - 3\right)^{5} .$

Use the Binomial Theorem to expand $\left(y - 1\right)^{6} .$

Use the Binomial Theorem to expand $\left(3 x - 2 y\right)^{5} .$

Use the Binomial Theorem to expand $\left(4 x - 3 y\right)^{4} .$

Find the third term of $\left(x + y\right)^{6} .$

Find the fifth term of $\left(a + b\right)^{8} .$

Find the coefficient of the $x^{5}$ term of $\left(x + 4\right)^{8} .$

Find the coefficient of the $x^{4}$ term of $\left(x + 2\right)^{7} .$

Expand each binomial using Pascal’s Triangle.

$\left(a + b\right)^{8}$

Expand each binomial using Pascal’s Triangle.

$\left(p + q\right)^{9}$

Expand each binomial using Pascal’s Triangle.

$\left(a - b\right)^{6}$

Expand each binomial using Pascal’s Triangle.

$\left(x + 5\right)^{3}$

Expand each binomial using Pascal’s Triangle.

$\left(y + 1\right)^{7}$

Expand each binomial using Pascal’s Triangle.

$\left(z - 2\right)^{6}$

Expand each binomial using Pascal’s Triangle.

$\left(3 x - 1\right)^{5}$

Expand each binomial using Pascal’s Triangle.

$\left(3 x - 5\right)^{3}$

Expand each binomial using Pascal’s Triangle.

$\left(3 x + 5 y\right)^{3}$

Evaluate.

ⓐ $\begin{pmatrix} 7 \\ 1 \end{pmatrix}$ ⓑ $\begin{pmatrix} 4 \\ 4 \end{pmatrix}$ ⓒ $\begin{pmatrix} 3 \\ 0 \end{pmatrix}$ ⓓ $\begin{pmatrix} 10 \\ 8 \end{pmatrix}$

Evaluate.

ⓐ $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ ⓑ $\begin{pmatrix} 5 \\ 5 \end{pmatrix}$ ⓒ $\begin{pmatrix} 8 \\ 0 \end{pmatrix}$ ⓓ $\begin{pmatrix} 11 \\ 9 \end{pmatrix}$

Expand each binomial.

$\left(m + n\right)^{5}$

Expand each binomial.

$\left(s + t\right)^{7}$

Expand each binomial.

$\left(y - 3\right)^{4}$

Expand each binomial.

$\left(q - 4\right)^{3}$

Expand each binomial.

$\left(5 x - 2 y\right)^{4}$

Expand each binomial.

$\left(3 x + 4 y\right)^{5}$

Find the indicated term in the expansion of the binomial.

Fifth term of $\left(a + b\right)^{9}$

Find the indicated term in the expansion of the binomial.

Seventh term of $\left(x - y\right)^{11}$

Find the coefficient of the indicated term in the expansion of the binomial.

$x^{6}$ term of $\left(x + 2\right)^{8}$

Find the coefficient of the indicated term in the expansion of the binomial.

$x^{7}$ term of $\left(x - 3\right)^{9}$

Find the coefficient of the indicated term in the expansion of the binomial.

$p^{5} q^{4}$ term of $\left(3 p + q\right)^{9}$

In your own words, explain the pattern of exponents for each variable in the expansion of.

In your own words, explain how to find a specific term in the expansion of a binomial without expanding the whole thing. Use an example to help explain.

Write the first five terms of the sequence whose general term is given.

$a_{n} = 3^{n} + 4$

Write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{2 n + 1}{4^{n}}$

Find a general term for the sequence whose first five terms are shown.

$9 , 18 , 27 , 36 , 45 , \ldots$

Find a general term for the sequence whose first five terms are shown.

$\frac{1}{e^{3}} , \frac{1}{e^{2}} , \frac{1}{e} , 1 , e , \ldots$

Find a general term for the sequence whose first five terms are shown.

$- \frac{1}{3} , - \frac{1}{2} , - \frac{3}{5} , - \frac{2}{3} , - \frac{5}{7} , \ldots$

Using factorial notation, write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{n !}{\left(n + 2\right) !}$

Expand the partial sum and find its value.

$\sum_{i = 1}^{7} \left(\right. 2 i - 5 \left.\right)$

Expand the partial sum and find its value.

$\sum_{k = 0}^{4} \frac{4}{k !}$

Write each sum using summation notation.

$- \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \frac{1}{81} - \frac{1}{243}$

Write each sum using summation notation.

$4 + 2 + \frac{4}{3} + 1 + \frac{4}{5}$

Determine if each sequence is arithmetic, and if so, indicate the common difference.

$−7 , −1 , 5 , 11 , 17 , 23 , \ldots$

Write the first five terms of each arithmetic sequence with the given first term and common difference.

$a_{1} = 5$ and $d = 3$

Write the first five terms of each arithmetic sequence with the given first term and common difference.

$a_{1} = −13$ and $d = 6$

Find the term described using the information provided.

Find the thirtieth term of a sequence where the first term is 16 and the common difference is $−5 .$

Find the indicated term and give the formula for the general term.

Find the eighteenth term of a sequence where the fifth term is $12$ and the common difference is seven.

Find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

The fifth term is 17 and the fourteenth term is 53.

Find the sum of the first 30 terms of each arithmetic sequence.

$7 , 4 , 1 , −2 , −5 , \ldots$

Find the sum of the first fifteen terms of the arithmetic sequence whose general term is given.

$a_{n} = 4 n + 7$

Find each sum.

$\sum_{i = 1}^{50} \left(\right. 4 i - 5 \left.\right)$

Find each sum.

$\sum_{i = 1}^{35} \left(\right. i + 10 \left.\right)$

Determine if the sequence is geometric, and if so, indicate the common ratio.

$5 , 10 , 15 , 20 , 25 , 30 , \ldots$

Determine if the sequence is geometric, and if so, indicate the common ratio.

$9 , −18 , 36 , −72 , 144 , −288 , \ldots$

Write the first five terms of each geometric sequence with the given first term and common ratio.

$a_{1} = 128$ and $r = \frac{1}{4}$

Find the indicated term of a sequence where the first term and the common ratio is given.

Find $a_{9}$ given $a_{1} = 6$ and $r = 2 .$

Find the indicated term of the given sequence. Find the general term of the sequence.

Find $a_{12}$ of the sequence, $6 , −24 , 96 , −384 , 1536 , −6144 , \ldots$

Find the sum of the first fifteen terms of each geometric sequence.

$−4 , 8 , −16 , 32 , −64 , 128 \ldots$

Find the sum of the first fifteen terms of each geometric sequence.

$3125 , 625 , 125 , 25 , 5 , 1 \ldots$

Find the sum

$\sum_{i = 1}^{6} 24 \left(\frac{1}{2}\right)^{i}$

Find the sum of each infinite geometric series.

$49 + 7 + 1 + \frac{1}{7} + \frac{1}{49} + \frac{1}{343} + \ldots$

Write each repeating decimal as a fraction.

$0. \overset{—}{36}$

Solve the problem.

Adam just got his first full-time job after graduating from high school at age 17. He decided to invest $\$ 300$ per month in an IRA (an annuity). The interest on the annuity is $7 \%$ which is compounded monthly. How much will be in Adam’s account when he retires at his sixty-seventh birthday?

Expand each binomial using Pascal’s Triangle.

$\left(x - y\right)^{4}$

Expand each binomial using Pascal’s Triangle.

$\left(2 y - 3\right)^{5}$

Evaluate.

ⓐ $\begin{pmatrix} 11 \\ 1 \end{pmatrix}$
ⓑ $\begin{pmatrix} 12 \\ 12 \end{pmatrix}$
ⓒ $\begin{pmatrix} 13 \\ 0 \end{pmatrix}$
ⓓ $\begin{pmatrix} 8 \\ 3 \end{pmatrix}$

Evaluate.

ⓐ $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$
ⓑ $\begin{pmatrix} 15 \\ 15 \end{pmatrix}$
ⓒ $\begin{pmatrix} 4 \\ 0 \end{pmatrix}$
ⓓ $\begin{pmatrix} 11 \\ 2 \end{pmatrix}$

Expand each binomial, using the Binomial Theorem.

$\left(t - 1\right)^{9}$

Expand each binomial, using the Binomial Theorem.

$\left(4 x + 3 y\right)^{4}$

Find the indicated term in the expansion of the binomial.

Seventh term of $\left(a + b\right)^{9}$

Find the coefficient of the indicated term in the expansion of the binomial.

$y^{4}$ term of $\left(y + 3\right)^{6}$

Find the coefficient of the indicated term in the expansion of the binomial.

$a^{3} b^{4}$ term of $\left(2 a + b\right)^{7}$

Write the first five terms of the sequence whose general term is given.

$a_{n} = \frac{\left(n + 2\right) !}{\left(n + 3\right) !}$

Write the first five terms of the sequence whose general term is given.

Expand the partial sum and find its value. $\sum_{i = 1}^{4} \left(−4\right)^{i}$

Write the first five terms of the sequence whose general term is given.

Write the first five terms of the arithmetic sequence with the given first term and common difference. $a_{1} = −13$ and $d = 3$

Write the first five terms of the sequence whose general term is given.

Find the twenty-third term of an arithmetic sequence whose seventh term is $11$ and common difference is three. Then find a formula for the general term.

Write the first five terms of the sequence whose general term is given.

Find the sum of the first 25 terms of the arithmetic sequence, $5 , 9 , 13 , 17 , 21 , \ldots$

Write the first five terms of the sequence whose general term is given.

Find the sum. $\sum_{i = 1}^{40} \left(\right. 5 i - 21 \left.\right)$

Determine if the sequence is arithmetic, geometric, or neither. If arithmetic, then find the common difference. If geometric, then find the common ratio.

$324 , 108 , 36 , 12 , 4 , \frac{4}{3} , \ldots$

Determine if the sequence is arithmetic, geometric, or neither. If arithmetic, then find the common difference. If geometric, then find the common ratio.

In the geometric sequence whose first term and common ratio are $a_{1} = 5$ and $r = 4 ,$ find $a_{11} .$

Determine if the sequence is arithmetic, geometric, or neither. If arithmetic, then find the common difference. If geometric, then find the common ratio.

Find the sum of the first thirteen terms of the geometric sequence, $2 , −6 , 18 , −54 , 162 , −486 \ldots$

Find the sum.

$1 - \frac{1}{5} + \frac{1}{25} - \frac{1}{125} + \frac{1}{625} - \frac{1}{3125} + \ldots$

Find the sum.

Dave just got his first full-time job after graduating from high school at age 18. He decided to invest $450 per month in an IRA (an annuity). The interest on the annuity is 6% which is compounded monthly. How much will be in Adam’s account when he retires at his sixty-fifth birthday?

Find the sum.

Evaluate each binomial coefficient. ⓐ $\begin{pmatrix} 8 \\ 1 \end{pmatrix}$
ⓑ $\begin{pmatrix} 16 \\ 16 \end{pmatrix}$ ⓒ $\begin{pmatrix} 12 \\ 0 \end{pmatrix}$ ⓓ $\begin{pmatrix} 10 \\ 6 \end{pmatrix}$