Find the HCF and LCM of:
1 + 4x + 4x^2 - 16x^4.. ,1 + 2x - 8x^3 - 16x^4.., 16x^4 + 4x^2 + 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).
To find the highest common factor (HCF) and lowest common multiple (LCM) of the given polynomials, we need to factor each polynomial and then find the product of common factors for the HCF and the product of all factors for the LCM.
The given polynomials are:
1 + 4x + 4x^2 - 16x^4
1 + 2x - 8x^3 - 16x^4
16x^4 + 4x^2 + 1
Let's factor each polynomial:
For the first polynomial, 1 + 4x + 4x^2 - 16x^4:
= 1 - 16x^4 + 4x + 4x^2
= (1 - 4x)(1 + 4x - 4x^2)
For the second polynomial, 1 + 2x - 8x^3 - 16x^4:
= 1 - 16x^4 + 2x - 8x^3
= (1 - 4x)(1 + 4x + 2x^2 + 4x^3)
For the third polynomial, 16x^4 + 4x^2 + 1:
= (4x^2 + 1)^2
Now, let's determine the common factors for the HCF by comparing the factors of each polynomial:
The common factor is (1 - 4x).
Next, let's find the product of the common factors for the HCF:
HCF = (1 - 4x)
Now, let's determine the factors for the LCM by multiplying all unique factors from each polynomial:
The factors for the LCM are (1 - 4x)(1 + 4x - 4x^2)(1 + 4x + 2x^2 + 4x^3)(4x^2 + 1)^2
Finally, let's find the LCM by multiplying all the factors together:
LCM = (1 - 4x)(1 + 4x - 4x^2)(1 + 4x + 2x^2 + 4x^3)(4x^2 + 1)^2
So, the HCF is (1 - 4x) and the LCM is (1 - 4x)(1 + 4x - 4x^2)(1 + 4x + 2x^2 + 4x^3)(4x^2 + 1)^2.
To find the highest common factor (HCF) and lowest common multiple (LCM) of the given polynomials, first, we need to factorize each polynomial completely.
The given polynomials are:
P1(x) = 1 + 4x + 4x^2 - 16x^4
P2(x) = 1 + 2x - 8x^3 - 16x^4
P3(x) = 16x^4 + 4x^2 + 1
To factorize each polynomial, we can look for common factors among the terms. Let's factorize each polynomial one by one.
Factorizing P1(x):
P1(x) = 1 + 4x + 4x^2 - 16x^4
= (1 - 4x^4) + 4x(1 + x)
Factorizing P2(x):
P2(x) = 1 + 2x - 8x^3 - 16x^4
= (1 - 8x^4) + 2x(1 - 4x^2)
Factorizing P3(x):
P3(x) = 16x^4 + 4x^2 + 1
= (4x^2 + 1)^2
Now that we have factorized each polynomial, we can proceed to find the HCF and LCM.
HCF:
The HCF of the given polynomials can be found by taking common factors raised to their smallest powers.
The common factors among the factorized polynomials are (1 - 4x^4) and (1 + 4x), as they are present in all polynomials.
So, the HCF of the given polynomials is:
HCF(x) = (1 - 4x^4) * (1 + 4x)
LCM:
The LCM of the given polynomials can be found by multiplying all unique factors raised to their highest powers.
The factors in P1(x) are (1 - 4x^4) and (1 + 4x).
The factors in P2(x) are (1 - 8x^4) and (1 - 4x^2).
The factors in P3(x) are (4x^2 + 1).
Taking all the unique factors and their highest powers, we get the LCM as follows:
LCM(x) = (1 - 4x^4) * (1 + 4x) * (1 - 8x^4) * (1 - 4x^2) * (4x^2 + 1)^2
Therefore, the HCF and LCM of the given polynomials are:
HCF(x) = (1 - 4x^4) * (1 + 4x)
LCM(x) = (1 - 4x^4) * (1 + 4x) * (1 - 8x^4) * (1 - 4x^2) * (4x^2 + 1)^2