Chapter 8.7, Problem 5CYU

To determine

To find: That the given polynomial is a difference of squares, if yes then factor it.

Answer to Problem 5CYU

The polynomial $16m^2-k^4$ is a difference of two squares, so it is factorable. The factored polynomial is $16m^2-k^4=\left(4m-k^2\right)\left(4m+k^2\right)$ .

Explanation of Solution

Given information:

The given polynomial is $16m^2-k^4$ .

Concept used:

The difference of two squares can be factored as $a^2-b^2=\left(a-b\right)\left(a+b\right)$ .

Calculations:

The given polynomial P (say) is $P=16m^2-k^4$ .

The polynomial can be written as $P={\left(4m\right)}^2-{\left(k^2\right)}^2$ , i.e., difference of two squares.

Since, the given polynomial $16m^2-k^4$ is a difference of two squares, so it can be factorized.

The factors of polynomial P are

  $P={\left(4m\right)}^2-{\left(k^2\right)}^2$

  $\Rightarrow P=\left(4m-k^2\right)\left(4m+k^2\right)$

Thus, the given polynomial $16m^2-k^4$ is a difference of two squares, it can be factorized. The factored polynomial is $16m^2-k^4=\left(4m-k^2\right)\left(4m+k^2\right)$ .

Conclusion:

The polynomial $16m^2-k^4$ is a difference of two squares, so it is factorable. The factored polynomial is $16m^2-k^4=\left(4m-k^2\right)\left(4m+k^2\right)$ .