Problem 5 - Algebra and Trigonometry

Question

Factor out the greatest common factor. $$9 x^{4}-18 x^{3}+27 x^{2}$$

Step-by-Step Solution

Answer

The greatest common factor of the expression \(9x^{4} - 18x^{3} + 27x^{2}\) is \(9x^{2}\), so factored out it becomes: \(9x^{2}(x^{2} - 2x + 3)\).

Step 1: Identify the greatest common factor

The greatest common factor is the highest number or term that divides each term of the expression evenly. Here, the coefficients are 9, 18 and 27, and the variables are \(x^{4}\), \(x^{3}\) and \(x^{2}\). The greatest common factor of the coefficients 9,18,27 is 9. The greatest common factor of the powers of \(x\) is \(x^{2}\), because it is the highest power that each term of \(x\) can be divided by without a remainder.

Step 2: Divide each term by the greatest common factor

Divide each term in the expression \(9x^{4} - 18x^{3} + 27x^{2}\) by the greatest common factor which is \(9x^{2}\). \((9x^{4} / 9x^{2}) - (18x^{3} / 9x^{2}) + (27x^{2} / 9x^{2})\) gives \(x^{2} - 2x + 3\).

Step 3: Write the final factored expression

The final expression is the greatest common factor multiplied by the result from step 2. This gives us: \(9x^{2}(x^{2} - 2x + 3)\).