Question

Find the HCF and LCM of:
1 + 4x + 4x^2 - 16x^4.. ,1 + 2x - 8x^3 - 16x^4.., 16x^4 + 4x^2 + 1

Answer 1

To find the HCF (highest common factor) and LCM (lowest common multiple) of the given polynomials, we first have to factorize each polynomial, if possible.

First, notice that the given polynomials are:
P1 = 1 + 4x + 4x^2 - 16x^4
P2 = 1 + 2x - 8x^3 - 16x^4
P3 = 16x^4 + 4x^2 + 1

Let's find the factors of each polynomial.

P1: 1 + 4x + 4x^2 - 16x^4 = (1 + 2x)^2 - (4x^2)^2 = (1 + 2x - 4x^2)(1 + 2x + 4x^2)

P2: 1 + 2x - 8x^3 - 16x^4 = (1 + 2x)(1 - 4x^2 - 8x^3)

P3: 16x^4 + 4x^2 + 1 = (4x^2 + 1)^2

Now let's find the HCF of the three polynomials.

The HCF is the product of the common factors with the lowest power. In this case, the common factor in all three polynomials is 1.

HCF = 1

Now let's find the LCM.

The LCM is the product of all the factors with the highest power. In this case,

LCM = (1 + 2x)(1 + 2x + 4x^2)(1 - 4x^2 - 8x^3)(4x^2 + 1)

So, the HCF is 1 and the LCM is (1 + 2x)(1 + 2x + 4x^2)(1 - 4x^2 - 8x^3)(4x^2 + 1).