Question

Factor the expression completely. Begin by factoring out the lowest power of each common factor.

${\mathbf{x}}^{5/2}\mathbf{-}{\mathbf{x}}^{1/2}$

Short answer

The solution$x^{5/2}-x^{1/2}=x^{1/2}\left(x+1\right)\left(x-1\right)$

Step-by-step solution

Step 1. Apply the concept of distributive property.

If A, B and C are real numbers, then

$A\left(B+C\right)=\left(B+C\right)A=AB+AC$

Step 2. Apply the special product formula of “product of the sum and difference of terms”.

If A and B are any real numbers or algebraic expressions, then

$\left(A+B\right)\left(A-B\right)=A^2-B^2$

Step 3. Analyze the expression.

Given$x^{5/2}-x^{1/2}$

Let us consider first two terms,$x^{5/2},x^{1/2}$

1, 1 have greatest common factor 1

$x^{5/2},x^{1/2}$have greatest common factor$x^{1/2}$

So, the greatest common factor of the two terms is$x^{1/2}$

Step 4. Factor the polynomial.

Now, using the distributive property the polynomial can be factored as:

$\begin{array}{l}{x}^{5/2}-x^{1/2}=x^{1/2}\left(x^{5/2-1/2}-x^{1/2-1/2}\right)\mathrm{................}\left[\frac{x^a}{x^b}=x^{a-b}\right]\\ \Rightarrow x^{5/2}-x^{1/2}=x^{1/2}\left(x^2-1\right)\\ \Rightarrow x^{5/2}-x^{1/2}=x^{1/2}\left(x^2-1^2\right)\\ \Rightarrow x^{5/2}-x^{1/2}=x^{1/2}\left(x+1\right)\left(x-1\right)\mathrm{..............}\left[x^2-a^2=\left(x+a\right)\left(x-a\right)\right]\end{array}$