Question

Factorize the expression and divide them as directed:
$(3x^4-1875)÷(3x^2-75)$

Answer 1

Step 1: Factorize the expression

Numerator: $3x^4-1875$

$=3[3x^4-625]$

$=3[{\left(x^2\right)}^2-{\left(25\right)}^2][\because a^2-b^2=(a+b)(a-b)]$

$=3\left[(x^2-25)(x^2+25)\right]$

$=3[x^2-5^2(x^2+25)]$

$[\because a^2+b^2=(a+b)(a-b)]$

$=3\left[(x+5)(x-5)(x^2+25)\right]$

Denominator: $3x^2-75$

$=3[x^2-25]$

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

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


Step2: Perform division of given expression

$\frac{3x^4-1875}{3x^2-75}=\frac{3(x+5)(x-5)(x^2+25)}{3(x-5)(x+5)}$

$=x^2+25$

$\therefore (3x^4-1875)÷(3x^2-75)=x^2+25.$