Chapter 13, Problem 5CRT

a.

To determine

To expand: the expression ${\left(2x-\frac{1}{2}\right)}^5$ using binomial theorem.

Answer to Problem 5CRT

The expand expression is $32x^5+40x^4+20x^3+5x^2+\frac{5}{8}x+\frac{1}{32}$.

Explanation of Solution

Given information:

The expression is ${\left(2x-\frac{1}{2}\right)}^5$.

Calculation: to find the expand of the expression, use the formula,

  $\begin{array}{l}{\left(2x-\frac{1}{2}\right)}^5=C_0^5{\left(2x\right)}^5{\left(\frac{1}{2}\right)}^0+C_1^5{\left(2x\right)}^4{\left(\frac{1}{2}\right)}^1+C_2^5{\left(2x\right)}^3{\left(\frac{1}{2}\right)}^2+C_3^5{\left(2x\right)}^2{\left(\frac{1}{2}\right)}^3+C_4^5{\left(2x\right)}^1{\left(\frac{1}{2}\right)}^4+C_5^5{\left(2x\right)}^0{\left(\frac{1}{2}\right)}^5\\ {\left(2x-\frac{1}{2}\right)}^5=32x^5+40x^4+20x^3+5x^2+\frac{5}{8}x+\frac{1}{32}\end{array}$

b.

To determine

To find: the term $x^4$ in the expand expression ${\left(2x-\frac{1}{2}\right)}^{12}$ using binomial theorem.

Answer to Problem 5CRT

The term is $\frac{495}{16}{x}^4$.

Explanation of Solution

Given information:

The expression is ${\left(2x-\frac{1}{2}\right)}^{12}$.

Calculation: to find the expand of the expression, use the formula,

The $nth$ term in the binomial theorem,

  $T_r=C_{n-r}^n{\left(a\right)}^{n-r}{\left(b\right)}^r$

Now substitute and simplify,

  $\begin{array}{l}{T}_r=C_{n-r}^n{\left(a\right)}^{n-r}{\left(b\right)}^r\\ T_8=C_{12-8}^{12}{\left(2x\right)}^{12-8}{\left(\frac{1}{2}\right)}^8\\ T_8=495\left(16x^4\right)\left(\frac{1}{256}\right)\text{ [simplify]}\\ T_8=\frac{495}{16}{x}^4\end{array}$