13.6 Binomial Theorem

Find each binomial coefficient.

1. ⓐ $\begin{pmatrix} 5 \\ 3 \end{pmatrix}$
2. ⓑ $\begin{pmatrix} 9 \\ 2 \end{pmatrix}$
3. ⓒ $\begin{pmatrix} 9 \\ 7 \end{pmatrix}$

Find each binomial coefficient.

1. ⓐ $\begin{pmatrix} 7 \\ 3 \end{pmatrix}$
2. ⓑ $\begin{pmatrix} 11 \\ 4 \end{pmatrix}$

Write in expanded form.

1. ⓐ $\left(x + y\right)^{5}$
2. ⓑ $\left(3 x - y\right)^{4}$

Write in expanded form.

1. ⓐ $\left(x - y\right)^{5}$
2. ⓑ $\left(2 x + 5 y\right)^{3}$

Find the tenth term of $\left(x + 2 y\right)^{16}$ without fully expanding the binomial.

Find the sixth term of $\left(3 x - y\right)^{9}$ without fully expanding the binomial.

What is a binomial coefficient, and how it is calculated?

What is the Binomial Theorem and what is its use?

Evaluate the binomial coefficient.

$\begin{pmatrix} 6 \\ 2 \end{pmatrix}$

Evaluate the binomial coefficient.

$\begin{pmatrix} 7 \\ 4 \end{pmatrix}$

Evaluate the binomial coefficient.

$\begin{pmatrix} 10 \\ 9 \end{pmatrix}$

Evaluate the binomial coefficient.

$\begin{pmatrix} 17 \\ 6 \end{pmatrix}$

Use the Binomial Theorem to expand each binomial.

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

Use the Binomial Theorem to expand each binomial.

$\left(3 a + 2 b\right)^{3}$

Use the Binomial Theorem to expand each binomial.

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

Use the Binomial Theorem to expand each binomial.

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

Use the Binomial Theorem to expand each binomial.

$\left(x^{- 1} + 2 y^{- 1}\right)^{4}$

Use the Binomial Theorem to write the first three terms of each binomial.

$\left(a + b\right)^{17}$

Use the Binomial Theorem to write the first three terms of each binomial.

$\left(a - 2 b\right)^{15}$

Use the Binomial Theorem to write the first three terms of each binomial.

$\left(3 a + b\right)^{20}$

Use the Binomial Theorem to write the first three terms of each binomial.

$\left(x^{3} - \sqrt{y}\right)^{8}$

Find the indicated term of each binomial without fully expanding the binomial.

The fourth term of $\left(3 x - 2 y\right)^{5}$

Find the indicated term of each binomial without fully expanding the binomial.

The eighth term of $\left(7 + 5 y\right)^{14}$

Find the indicated term of each binomial without fully expanding the binomial.

The fifth term of $\left(x - y\right)^{7}$

Find the indicated term of each binomial without fully expanding the binomial.

The ninth term of $\left(a - 3 b^{2}\right)^{11}$

Find the indicated term of each binomial without fully expanding the binomial.

The eighth term of $\left(\frac{y}{2} + \frac{2}{x}\right)^{9}$

In the expansion of $\left(5 x + 3 y\right)^{n} ,$ each term has the form $\begin{pmatrix} n \\ k \end{pmatrix} a^{n – k} b^{k}$, where $k$ successively takes on the value $0 , 1 , 2 , ... , n .$ If $\begin{pmatrix} n \\ k \end{pmatrix} = \begin{pmatrix} 7 \\ 2 \end{pmatrix} ,$ what is the corresponding term?

Consider the expansion of $\left(x + b\right)^{40} .$ What is the exponent of $b$ in the $k \text{th}$ term?

Which expression cannot be expanded using the Binomial Theorem? Explain.

• $\left(x^{2} - 2 x + 1\right)$
• $\left(\sqrt{a} + 4 \sqrt{a} - 5\right)^{8}$
• $\left(x^{3} + 2 y^{2} - z\right)^{5}$
• $\left(3 x^{2} - \sqrt{2 y^{3}}\right)^{12}$