Factor the Polynomials?

Question

Factor Each Expression
16x^3-12x^2+20x-15

6x^3+12x^2+4x-8

Answer 1

1) The cubic polynomial is  16x³ - 12x² + 20x - 15

= 16x³ + 20x - 12x² - 15

= 4x(4x² + 5) - 3(4x² + 5)

Factoring of  16x³ - 12x² + 20x - 15 = (4x² + 5)(4x - 3).

Answer 2

2) The cubic polynomial is 6x³ + 12x² + 4x - 15

From factor theorem,

When f(c) = 0, then x - c is a factor of a polynomial.

Now find zeros of polynomial.

Graph

The points where it crosses the x axis  will give solutions to the polynomial function .

The graph crosses the x - axis at a point that would suggest a factor.

It crosses the x - axis at one point hence there are one real root.

x  = 0.5993

Use synthetic division to determine if the given value of x is a root of the polynomial.

Since f(0.5993) = 0, x = 0.5993 is a zero.

The depressed polynomial is  6x² + 15.5958x + 13.3465

Since the depressed polynomial of this zero, 6x² + 15.5958x + 13.3465 = 0, is quadratic,

Use the Quadratic Formula to find the roots of the related quadratic equation ax² + bx + c = 0

a = 6, b = 15.5958, c = 13.3465

The function as one real zero at x = 0.5993 and two imaginary solutions at x = - 1.299 + 0.73i , x = -1.299 - 0.73i.

6x³ + 12x² + 4x - 15 = [x - 0.5993][x - (- 1.299+0.73i)][(- 1.299 - 0.73i)].