Chapter 8.7, Problem 36PPS

To determine

To factor: The given polynomial, if possible. If the polynomial cannot be factored, write prime.

Answer to Problem 36PPS

The factored polynomial is $2a^2{b}^2-2a^2-2a{b}^3+2ab=2a\left(b-1\right)\left(b+1\right)\left(a-b\right)$ .

Explanation of Solution

Given information:

The given polynomial is $2a^2{b}^2-2a^2-2a{b}^3+2ab$ .

Concept used:

The polynomial of the form $a{c}^3+b{x}^2+cx+d$ are factored by grouping the terms having one or more common factor(s).

Calculations:

The given polynomial E (say) is $E=2a^2{b}^2-2a^2-2a{b}^3+2ab$ that can be written as

  $E=2a\left(a{b}^2-a-b^3+b\right)$

  $\Rightarrow E=2a\left(\left(a{b}^2-a\right)-\left(b^3-b\right)\right)$

  $\Rightarrow E=2a\left(a\left(b^2-1\right)-b\left(b^2-1\right)\right)$    Factor $\left(b^2-1\right)$ is common.

  $\Rightarrow E=2a\left(b^2-1\right)\left(a-b\right)$

  $\Rightarrow E=2a\left(b-1\right)\left(b+1\right)\left(a-b\right)$    Difference of two squares

Thus, the factored polynomial is $2a^2{b}^2-2a^2-2a{b}^3+2ab=2a\left(b-1\right)\left(b+1\right)\left(a-b\right)$ .

The given polynomial is not a prime polynomial.

Conclusion:

The factored polynomial is $2a^2{b}^2-2a^2-2a{b}^3+2ab=2a\left(b-1\right)\left(b+1\right)\left(a-b\right)$ .