Question

If $HCF$ & $LCM$ of two quadratic expression are $(x-5)$ and $x^3-19x-30$ then find the two expressions.

• A. $p\left(x\right)=(x-5)(x+2)andq\left(x\right)=(x-5)(x+3)$
• B. $p\left(x\right)=(x-2)(x-8)andq\left(x\right)=(x-1)(x-3)$
• C. $p\left(x\right)=(x+1)(x-2)andq\left(x\right)=(x-1)(x-3)$
• D. $p\left(x\right)=(x-1)(x-2)andq\left(x\right)=(x-1)(x-8)$

Answer 1

The correct option is A $p\left(x\right)=(x-5)(x+2)andq\left(x\right)=(x-5)(x+3)$

Given

$HCL=(x-5)$

$LCM=x^3-19x-30$

Let the two polynomials

$p\left(x\right)8q\left(x\right)$

Finding the factor of $LCM$

$x^3-19x-30$

$=(x-5)(x+2)(x+3)$

As $p\left(x\right)$ & $q\left(x\right)$, both are quadratic polynomial so, and product of polynomials $=LCM\times HCF$

So,

$p\left(x\right)\times q\left(x\right)=(x-5)(x-5)(x+2)(x+3)$

So,

$p\left(x\right)=(x-5)(x+2)$

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