Question

H.c.f of x^2+4x-12, x^3+6x^2-16x

Answer 1

Concept:

We will require the concept of H.C.F to solve this question

H.C.F is the highest common factor between two or more integers

Given:

We are given two polynomials:

x²+4x-12 and x³+6x²-16x

To find:

We are asked to find the H.C.F of the two polynomials given

Solution:

We will factorise both the polynomials first

the first polynomial:

x²+4x-12

=x²+6x-2x-12

=x(x+6)-2(x+6)

=(x+6)(x-2)..........................(1)

The second polynomial:

x³+6x²-16x

=x(x²+6x-16)

=x(x²+8x-2x-16)

=x{x(x+8)-2(x+8)}

=x(x+8)(x-2)...........................(2)

The two polynomials are factorized and we can see that there exists only one common factor between the two polynomials. So, this itself will be the highest common factor.

Thus,

From the two Eqns(1) and (2) we get that:

(x-2) is the highest common factor(H.C.F) of the two given polynomials.

Answer 2

Answer:

HCF of $x^2+4x-12$ and $x^3+6x^2-16x$ is (x-2)

Step-by-step explanation:

Given,

Two polynomials $x^2+4x-12$ and $x^3+6x^2-16x$

Required to find the H.C. F of these two polynomials

To find the HCF of these two polynomials, we have to factorize these polynomials

To factorize  $x^2+4x-12$ by splitting the middle term

To split the middle term, we need to find two numbers such that their sum is +4 and the product is -12

Two such numbers are +6 and -2

So, $x^2+4x-12$, can be written as

$x^2+4x-12 = x^2+6x-2x-12$

=x(x+6)-2(x+6)

=(x-2)(x+6)

Hence the factors of $x^2+4x-12$ are (x-2) and (x+6)

To Factorize $x^3+6x^2-16x$

$x^3+6x^2-16x = x(x^2 +6x-16)$

To split the middle term of $x^2 +6x-16$, we need to find two numbers such that their sum is +6 and the product is -12

Two such numbers are +8 and -2

Hence, $x^3+6x^2-16x = x(x^2 +8x-2x -16)$

=x(x(x+8)-2(x+8))

=x(x+8)(x-2)

Hence the factors of $x^3+6x^2-16x$  are x,(x+8),(x-2)

So the common factor of $x^2+4x-12$ and $x^3+6x^2-16x$ is (x-2)

Hence the HCF of $x^2+4x-12$ and $x^3+6x^2-16x$ is (x-2)