Chapter 8.2, Problem 3P

a.

To determine

To find: The factored form of $9x^6+15x^4+12x^2$

Answer to Problem 3P

  $3x^2(3x^4+5x^2+4)$

Explanation of Solution

Given information:

The polynomial $9x^6+15x^4+12x^2$

Calculation:

The prime factors of the terms of the polynomial are,

  $\begin{array}{l}9x^6=3\cdot 3\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\\ 15x^4=3\cdot 5\cdot x\cdot x\cdot x\cdot x\\ 12x^2=2\cdot 2\cdot 3\cdot x\cdot x\end{array}$

Then, $\text{GCF}=3\cdot x\cdot x=3x^2$

  $\begin{array}{l}9x^6+15x^4+12x^2\\ =3x^2(3x^4)+3x^2(5x^2)+3x^2(4)\\ =3x^2(3x^4+5x^2+4)[\text{factoring out the GCF]}\end{array}$

Thus, $9x^6+15x^4+12x^2=3x^2(3x^4+5x^2+4)$

b.

To determine

To find: $-6x^4-18x^3-12x^2$ written as the product of a polynomial with positive coefficients and a monomial.

Answer to Problem 3P

  $-6x^2(x^2+3x+2)$

Explanation of Solution

Given information:

The polynomial $-6x^4-18x^3-12x^2$

Calculation:

The prime factors of the terms of the polynomial are,

  $\begin{array}{l}6x^4=2\cdot 3\cdot x\cdot x\cdot x\cdot x\\ 18x^3=2\cdot 3\cdot 3\cdot x\cdot x\cdot x\\ 12x^2=2\cdot 2\cdot 3\cdot x\cdot x\end{array}$

Then, $\text{GCF}=2\cdot 3\cdot x\cdot x=6x^2$

  $\begin{array}{l}-6x^4-18x^3-12x^2\\ =6x^2(-x^2)+6x^2(-3x)+6x^2(-2)\\ =6x^2(-x^2-3x-2)[\text{factoring out the GCF]}\\ \text{=}-6x^2(x^2+3x+2)[\text{taking out the negative sign]}\end{array}$

Thus, $-6x^4-18x^3-12x^2=-6x^2(x^2+3x+2)$ which is the product of $(x^2+3x+2)$ , a polynomial with positive coefficients and $-6x^2$ , a monomial.