Chapter 8.7, Problem 12CYU

To determine

To factor: The given polynomial if possible. If it cannot be factored then to write it a prime.

Answer to Problem 12CYU

The factored polynomial is $2d^4-32f^4=2\left(d-2f\right)\left(d+2f\right)\left(d^2+4f^2\right)$ .

Explanation of Solution

Given information:

The given polynomial is $2d^4-32f^4$ .

Concept used:

The difference of two squares can be factored as $a^2-b^2=\left(a-b\right)\left(a+b\right)$ , and the difference $x^4-y^4=\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)$ .

Calculations:

The given polynomial E (say) is $E=2d^4-32f^4$ that can be written as

  $E=2\left(d^4-16f^4\right)$

  $E=2\left\{{\left(d^2\right)}^2-{\left(4f^2\right)}^2\right\}$       Difference of two squares

  $\Rightarrow E=2\left(d^2-4f^2\right)\left(d^2+4f^2\right)$ .

  $\Rightarrow E=2\left({(d)}^2-{(2f)}^2\right)\left(d^2+4f^2\right)$       Again difference of two squares

  $\Rightarrow E=2\left(d-2f\right)\left(d+2f\right)\left(d^2+4f^2\right)$

Thus, the factored polynomial is $2d^4-32f^4=2\left(d-2f\right)\left(d+2f\right)\left(d^2+4f^2\right)$ .

(The polynomial $2d^4-32f^4$ is not a prime polynomial.)

Conclusion:

The factored polynomial is $2d^4-32f^4=2\left(d-2f\right)\left(d+2f\right)\left(d^2+4f^2\right)$ .