Key Concept: Factor Theorem
According to the Factor Theorem, if \((x-a)\) is a factor of a polynomial \(f(x)\), then substituting \(x=a\) into the polynomial will result in \(f(a)=0\). Since \((x-5)\) is the HCF (Highest Common Factor) of the given polynomials, it must be a factor of both.
Applying the Factor Theorem
We substitute \(x=5\) into both polynomials and set the result to zero:
- For the first polynomial \(f(x) = x^2-x-p\): Substitute \(x=5\): \(f(5) = 5^2 - 5 - p = 0\) \(25 - 5 - p = 0\) \(20 - p = 0\) \(p = 20\)
- For the second polynomial \(g(x) = x^2-qx-10\): Substitute \(x=5\): \(g(5) = 5^2 - q(5) - 10 = 0\) \(25 - 5q - 10 = 0\) \(15 - 5q = 0\) \(5q = 15\) \(q = \frac{15}{5}\) \(q = 3\)
Calculating \((p+q)\)
Now, we find the required value of \((p+q)\) using the values we found for \(p\) and \(q\):
\((p+q) = 20 + 3\) \((p+q) = 23\)Therefore, the value of \((p+q)\) is 23.
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?
