Question

Find the square root of following polynomial $(x^2-25)(x^2+8x+15)(x^2-2x-15)$

• A. $(x+5)(x-5)(x+3)$
• B. $(x+5)(x+5)(x+3)$
• C. $(x+5)(x-5)(x-3)$
• D. None of these

Answer 1

Answer: (A)

$(x+5)(x-5)(x+3)$

We find the square root of $(x^2-25)(x^2+8x+15)(x^2-2x-15)$ as shown below:

$=\sqrt{(x^2-25)(x^2+8x+15)(x^2-2x-15)}$

$=\sqrt{(x^2-5^2)(x^2+5x+3x+15)(x^2-5x+3x-15)}$

$=\sqrt{\left[\right(x-5\left)\right(x+5\left)\right]\left[x\right(x+5)+3(x+5\left)\right]\left[x\right(x-5)+3(x-5\left)\right]}$

$(\because a^2-b^2=(a-b\left)\right(a+b\left)\right)$

$=\sqrt{\left[\right(x-5\left)\right(x+5\left)\right]\left[\right(x+3\left)\right(x+5\left)\right]\left[\right(x+3\left)\right(x-5\left)\right]}$

$=\sqrt{(x-5)(x+5)(x+3)(x+5)(x+3)(x-5)}$

$=\sqrt{(x-5{)}^2(x+5{)}^2(x+3{)}^2}$

$=\sqrt{\left[\right(x-5\left)\right(x+5\left)\right(x+3){]}^2}$

$=(x-5)(x+5)(x+3)$

Hence, the square root of $(x^2-25)(x^2+8x+15)(x^2-2x-15)$ is $(x-5)(x+5)(x+3)$.