Question

Select the expression that is equivalent to the given polynomial.
16r4 – 625
O A. (2x + 5)(2x - 5)(2x + 5i) (2x – 5i)
B. (2x - 5)(2x – 5)(2x – 5i)(2x – 5i)
C. (2x - 5)(2x – 5)(2x + 5i)(2x – 5i)
D. (2x + 5)(2x + 5)(2x + 5) (21 + 5i)

Answer 1

Observe that $16r^4$ is the square of $4r^2$, and 625 is the square of 25.

So, your expression is a difference of squares, and as such we can rewrite it as

$16r^4-625=(4r^2+25)(4r^2-25)$

Now again, $4r^2$ is the square of $2r$, and 25 is the square of 5. So, we have a sum and a difference of squares.

But if we think of 25 as $-(5i)^2$, we have again the difference of two squares, so we have

$4r^2+25=(2r+5i)(2r-5i)$

$4r^2-25=(2r+5)(2r-5)$

Answer 2

Option A is the correct expression equivalent to 16r⁴-625.

The options in the question must have r in place of x.

What does the imaginary number i mean?

The imaginary number 'i' is the solution of the algebraic equation x²+1.

Thus, x²+1=(x+i)(x-i).

How to factorize (a²+b²)?

We can observe that (a+bi)(a-bi)=(a²+b²)+i(-ab+ba)=a²+b².

Hence the factorization of a²+b² is (a+bi)(a-bi)

How to factorize the given polynomial?

16r⁴-625

=(4r²-25)(4r²+25)

=(2r-5)(2r+5)(2r+5i)(2r-5i)

So, option A is the correct expression equivalent to 16r⁴-625.

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