Finding the HCF of Given Polynomials
The question asks for the Highest Common Factor (HCF) of three given polynomials:
- \(P_1 = x^4 - 13x^2y^2 - 300y^4\)
- \(P_2 = x^3 - 4x^2y - 4xy^2 - 5y^3\)
- \(P_3 = x^3 - 125y^3\)
To find the HCF, we need to factorize each polynomial completely and then identify the common factors.
Factorizing Polynomial 1: \(x^4 - 13x^2y^2 - 300y^4\)
This polynomial can be treated as a quadratic equation in terms of \(x^2\). Let \(u = x^2\). The expression becomes \(u^2 - 13uy^2 - 300y^4\). We need to find two terms whose product is \(-300y^4\) and whose sum is \(-13y^2\).
By trying factors of 300, we find that \(-25y^2\) and \(12y^2\) multiply to \(-300y^4\) and add up to \(-13y^2\).
So, we can rewrite the expression as:
\(u^2 - 25y^2u + 12y^2u - 300y^4\)
Factoring by grouping:
\(u(u - 25y^2) + 12y^2(u - 25y^2)\)
\((u + 12y^2)(u - 25y^2)\)
Substitute back \(u = x^2\):
\((x^2 + 12y^2)(x^2 - 25y^2)\)
The term \(x^2 - 25y^2\) is a difference of squares (\(a^2 - b^2 = (a-b)(a+b)\)), where \(a=x\) and \(b=5y\). So, \(x^2 - 25y^2 = (x - 5y)(x + 5y)\).
Therefore, the complete factorization of the first polynomial is:
\(P_1 = (x - 5y)(x + 5y)(x^2 + 12y^2)\)
Factorizing Polynomial 2: \(x^3 - 4x^2y - 4xy^2 - 5y^3\)
Let's test if \((x - 5y)\) is a factor by substituting \(x = 5y\) into the polynomial:
\((5y)^3 - 4(5y)^2y - 4(5y)y^2 - 5y^3\)
\(= 125y^3 - 4(25y^2)y - 20y^3 - 5y^3\)
\(= 125y^3 - 100y^3 - 20y^3 - 5y^3\)
\(= 125y^3 - 125y^3 = 0\)
Since the result is 0, \((x - 5y)\) is indeed a factor. Now, we perform polynomial division to find the other factor.
Dividing \(x^3 - 4x^2y - 4xy^2 - 5y^3\) by \((x - 5y)\):
x^2 + xy + y^2 |
____________________ |
x - 5y | x^3 - 4x^2y - 4xy^2 - 5y^3 |
-(x^3 - 5x^2y) |
____________________ |
x^2y - 4xy^2 |
-(x^2y - 5xy^2) |
____________________ |
xy^2 - 5y^3 |
-(xy^2 - 5y^3) |
_____________ |
0 |
Thus, the factorization is:
\(P_2 = (x - 5y)(x^2 + xy + y^2)\)
Factorizing Polynomial 3: \(x^3 - 125y^3\)
This is a difference of cubes, \(a^3 - b^3\), where \(a = x\) and \(b = 5y\). The formula for the difference of cubes is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
Applying the formula:
\(P_3 = (x - 5y)(x^2 + x(5y) + (5y)^2)\)
\(P_3 = (x - 5y)(x^2 + 5xy + 25y^2)\)
Identifying the Highest Common Factor (HCF)
Now, let's list the factors of each polynomial:
- \(P_1 = (x - 5y)(x + 5y)(x^2 + 12y^2)\)
- \(P_2 = (x - 5y)(x^2 + xy + y^2)\)
- \(P_3 = (x - 5y)(x^2 + 5xy + 25y^2)\)
The common factor present in all three polynomials is \((x - 5y)\).
Therefore, the HCF of the three given polynomials is \(x - 5y\).
Similar Questions
- If HCF of 768 and \(x^6y^2\) is \(32xy\) for natural numbers \(x \ge 2, y \ge 2\), then what is the value of \((x + y)\)?
- What is the HCF of \(2^{36}-1\) and \(2^{45}-1\)?
The HCF of \(x\) and \(y\) is \(H\). Consider the following statements in respect of the HCF of \(p=\frac{x^3 + y^3}{x^2-xy+y^2}\) and \(q=\frac{x^3-y^3}{x^2+xy+y^2}\) :
I. The HCF of \(p\) and \(q\) can be \(H\).
II. The HCF of \(p\) and \(q\) can be \(2H\).
Which of the statements given above is/are correct?
- What is the HCF of \(x^3 + y^3 + 3xy-1\) and \((x + y)^2 - 1\)?
- A number N is such that when divided by 4, 6, 7 or 9, it leaves 3 as remainder. What is the smallest 4-digit number that satisfies this property?
- A plank of wood 4·25 m long and 3·4 m wide is to be cut into square pieces of equal size. How many square pieces of largest size can be cut from the plank, if no wastage is allowed ?
- Let \(N\) be a 5-digit number. When \(N\) is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of \(N\)?
Let \(N\) be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of he following is correct?
Let \(n\) be a natural number. The HCF of \(n, n + 10\) is 10. If the LCM is \(x\) (a 2-digit number), then how many values of \(x\) are possible?
Consider the following statements in respect of prime numbers p and q:
- I. Their LCM is always an odd number.
- II. Sum of their LCM and HCF is always an even number.
Which of the statements given above is/are correct?
Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
What is the least number which when divided by 12,20 and 24 leaves in each case a remainder of 8?
Which of the following is a pair of co-primes?
