Question

Factor completely: \(30 x^{2}(x-7)^{3 / 2}+15 x^{3}(x-7)^{1 / 2}\).

Short answer

The completely factored form is \(15 x^{2} (x-7)^{1/2} (3x - 14)\).

Step-by-step solution

Identify Common Factors

First, look for common factors in the terms. Both terms, \(30 x^{2}(x-7)^{3 / 2}\) and \(15 x^{3}(x-7)^{1 / 2}\), share common factors. These factors include the numeric coefficients (30 and 15) and powers of \(x\) and \((x-7)\).

Factor Out Greatest Common Divisor (GCD)

The numeric GCD of 30 and 15 is 15. The greatest common factor involving \(x\) is \(x^2\) since it is the lowest power of \(x\) in both terms. The greatest common factor involving \((x-7)\) is \((x-7)^{1/2}\) since it is also the lowest power of \((x-7)\) in both terms. Factor these out:\[30 x^{2}(x-7)^{3/2} + 15 x^{3}(x-7)^{1/2} = 15 x^{2} (x-7)^{1/2} \bigg(2 (x-7) + x \bigg)\]

Simplify the Expression Inside the Parenthesis

Simplify the expression inside the parenthesis:\[\bigg(2 (x-7) + x \bigg) = 2x - 14 + x = 3x - 14\]Thus, the equation becomes:\[15 x^{2} (x-7)^{1/2} (3x - 14)\]

Write the Final Factored Form

The completely factored form of the expression is:\[15 x^{2} (x-7)^{1/2} (3x - 14)\]

Key concepts

Greatest Common Divisor

When you start factoring algebraic expressions, identifying the greatest common divisor (GCD) is crucial. The GCD is the highest number that divides exactly into two or more numbers. For algebraic terms, this includes coefficients and variable powers.

Let's look at our example: for the terms \(30 x^{2}(x-7)^{3/2}\) and \(15 x^{3}(x-7)^{1/2}\):

• Numeric coefficients: 30 and 15 share a GCD of 15.
• \