\(x^2-a^2\)
To solve this problem, we need to use the relationship between the Least Common Multiple (LCM), Highest Common Factor (HCF), and the given polynomials.
We are given:
- LCM of \(p(x)\) and \(q(x)\) is \((x+a)(x^3-a^3)\)
- HCF of \(p(x)\) and \(q(x)\) is \((x^2-ax+a^2)\)
- Polynomial \(p(x) = x^4 + a^2x^2 + a^4\)
And we need to find \(q(x)\).
Using the relationship:
\(LCM(p(x), q(x)) \cdot HCF(p(x), q(x)) = p(x) \cdot q(x).\)
Substitute the given values into the equation:
\(\left[(x+a)(x^3-a^3)\right] \cdot (x^2-ax+a^2) = (x^4 + a^2x^2 + a^4) \cdot q(x).\)
First, let's simplify \((x+a)(x^3-a^3)\):
- Recognize that \(x^3-a^3 = (x-a)(x^2+ax+a^2)\) (using the difference of cubes).
- So, \((x+a)(x^3-a^3) = (x+a)(x-a)(x^2+ax+a^2)\).
Thus, the LCM can be represented as:
\((x+a)(x-a)(x^2+ax+a^2).\)
Now the equation becomes:
\((x+a)(x-a)(x^2+ax+a^2)(x^2-ax+a^2) = (x^4 + a^2x^2 + a^4) \cdot q(x).\)
Simplify the left-hand side:
- \((x+a)(x-a) = x^2-a^2\)
- So, the left side is \((x^2-a^2)(x^2+ax+a^2)(x^2-ax+a^2)\).
With this simplification, we substitute back:
\((x^2-a^2) = q(x),\)
where the root of simplification and cancellation is observed. Thus, the correct polynomial \(q(x)\) that satisfies the equation is:
- \(q(x) = x^2-a^2\)
Therefore, the option that matches is \(x^2-a^2\).
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 ?
- What is the HCF of \(x^4 - 13x^2y^2 - 300y^4\), \(x^3 - 4x^2y - 4xy^2 - 5y^3\) and \(x^3 - 125y^3\)?
- 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?
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?
