Question

1. factor - completely polynomial 3x³ - 27x² + 60x 15. 2x⁵ - 72x³

Explanation:

Step1: Factor \(3x^3 - 27x^2 + 60x\)

First, find the greatest common factor (GCF) of the terms. The GCF of \(3x^3\), \(-27x^2\), and \(60x\) is \(3x\).
Divide each term by \(3x\):
\(\frac{3x^3}{3x} - \frac{27x^2}{3x} + \frac{60x}{3x} = x^2 - 9x + 20\)
Now factor the quadratic \(x^2 - 9x + 20\). We need two numbers that multiply to \(20\) and add to \(-9\), which are \(-4\) and \(-5\).
So \(x^2 - 9x + 20 = (x - 4)(x - 5)\)
Thus, \(3x^3 - 27x^2 + 60x = 3x(x - 4)(x - 5)\)

Step2: Factor \(2x^5 - 72x^3\)

First, find the GCF of the terms. The GCF of \(2x^5\) and \(-72x^3\) is \(2x^3\).
Divide each term by \(2x^3\):
\(\frac{2x^5}{2x^3} - \frac{72x^3}{2x^3} = x^2 - 36\)
Notice that \(x^2 - 36\) is a difference of squares, which factors as \((x - 6)(x + 6)\)
Thus, \(2x^5 - 72x^3 = 2x^3(x - 6)(x + 6)\)

Answer:

\(3x^3 - 27x^2 + 60x = \boldsymbol{3x(x - 4)(x - 5)}\)
\(2x^5 - 72x^3 = \boldsymbol{2x^3(x - 6)(x + 6)}\)