13.6 Binomial theorem  (Page 3/6)

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

What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?

What is the Binomial Theorem and what is its use?

When is it an advantage to use the Binomial Theorem? Explain.

$\left(\begin{array}{c}6\\ 2\end{array}\right)$

$\left(\begin{array}{c}5\\ 3\end{array}\right)$

$\left(\begin{array}{c}7\\ 4\end{array}\right)$

$\left(\begin{array}{c}9\\ 7\end{array}\right)$

$\left(\begin{array}{c}10\\ 9\end{array}\right)$

$\left(\begin{array}{c}25\\ 11\end{array}\right)$

$\left(\begin{array}{c}17\\ 6\end{array}\right)$

$\left(\begin{array}{c}200\\ 199\end{array}\right)$

${(4a-b)}^3$

${(5a+2)}^3$

${(3a+2b)}^3$

${(2x+3y)}^4$

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

${(3x-2y)}^4$

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

${\left(\frac{1}{x}+3y\right)}^5$

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

${(\sqrt{x}-\sqrt{y})}^5$

${(a+b)}^{17}$

${(x-1)}^{18}$

${(a-2b)}^{15}$

${(x-2y)}^8$

${(3a+b)}^{20}$

${(2a+4b)}^7$

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

The fourth term of ${(2x-3y)}^4$

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

The third term of ${(6x-3y)}^7$

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

The seventh term of ${(a+b)}^{11}$

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

The tenth term of ${(x-1)}^{12}$

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

The fourth term of ${\left(x^3-\frac{1}{2}\right)}^{10}$

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

Find and graph $f_1(x),$ such that $f_1(x)$ is the first term of the expansion.

Find and graph $f_2(x),$ such that $f_2(x)$ is the sum of the first two terms of the expansion.

Find and graph $f_3(x),$ such that $f_3(x)$ is the sum of the first three terms of the expansion.

Find and graph $f_4(x),$ such that $f_4(x)$ is the sum of the first four terms of the expansion.

Find and graph $f_5(x),$ such that $f_5(x)$ is the sum of the first five terms of the expansion.

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

In the expansion of ${\left(a+b\right)}^n,$ the coefficient of $a^{n-k}{b}^k$ is the same as the coefficient of which other term?

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

Find $\left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{c}n\\ k\end{array}\right)$ and write the answer as a binomial coefficient in the form $\left(\begin{array}{c}n\\ k\end{array}\right).$ Prove it. Hint: Use the fact that, for any integer $p,$ such that $p\ge 1,p!=p(p-1)!\text{.}$

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

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