Chapter 1.3, Problem 92E

To determine

To calculate: The factor of the expression ${\left(x-1\right)}^{\frac{7}{2}}-{\left(x-1\right)}^{\frac{3}{2}}$ .

Answer to Problem 92E

The factor of the expression ${\left(x-1\right)}^{\frac{7}{2}}-{\left(x-1\right)}^{\frac{3}{2}}$ is $x\left(x-2\right)\sqrt{{\left(x-1\right)}^3}$ .

Explanation of Solution

Given information:

The expression ${\left(x-1\right)}^{\frac{7}{2}}-{\left(x-1\right)}^{\frac{3}{2}}$ .

Formula used:

To factor an expression, split the terms in the expression into multiplication of simpler expressions, then take the common power out and group the expressions together.

The special factoring formula for perfect square which is mathematically expressed as,

  ${\left(A-B\right)}^2=A^2-2AB+B^2$

Calculation:

Consider the given expression ${\left(x-1\right)}^{\frac{7}{2}}-{\left(x-1\right)}^{\frac{3}{2}}$ .

Recall that to factor an expression, split the terms in the expression into multiplication of simpler expressions, then take the common power out and group the expressions together.

The greatest common factor of these terms is ${\left(x-1\right)}^{\frac{3}{2}}$ .

Apply it,

  $\begin{array}{c}{\left(x-1\right)}^{\frac{7}{2}}-{\left(x-1\right)}^{\frac{3}{2}}={\left(x-1\right)}^{\frac{3}{2}}\left[{\left(x-1\right)}^{\frac{7}{2}}\cdot {\left(x-1\right)}^{-\frac{3}{2}}-1\right]\\ ={\left(x-1\right)}^{\frac{3}{2}}\left[{\left(x-1\right)}^{\frac{7}{2}-\frac{3}{2}}-1\right]\\ ={\left(x-1\right)}^{\frac{3}{2}}\left[{\left(x-1\right)}^2-1\right]\end{array}$

Recall the special factoring formula for perfect square which is mathematically expressed as,

  ${\left(A-B\right)}^2=A^2-2AB+B^2$

So, ${\left(x-1\right)}^{\frac{3}{2}}\left[{\left(x-1\right)}^2-1\right]$ will be further simplified as,

  $\begin{array}{c}{\left(x-1\right)}^{\frac{7}{2}}-{\left(x-1\right)}^{\frac{3}{2}}={\left(x-1\right)}^{\frac{3}{2}}\left[{\left(x-1\right)}^2-1\right]\\ ={\left(x-1\right)}^{\frac{3}{2}}\left[{\left(x\right)}^2-2\left(x\right)\left(1\right)+1^2-1\right]\\ =\sqrt{{\left(x-1\right)}^3}\left[x^2-2x\right]\\ =\sqrt{{\left(x-1\right)}^3}\left[x\left(x-2\right)\right]\end{array}$

It can be rewritten as,

  ${\left(x-1\right)}^{\frac{7}{2}}-{\left(x-1\right)}^{\frac{3}{2}}=x\left(x-2\right)\sqrt{{\left(x-1\right)}^3}$

Thus, the factor of expression ${\left(x-1\right)}^{\frac{7}{2}}-{\left(x-1\right)}^{\frac{3}{2}}$ is $x\left(x-2\right)\sqrt{{\left(x-1\right)}^3}$ .