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.
Important Questions from LCM and HCF
The greatest three-digit number which is divisible by 14, 28, and 42 is:
What is the greatest number that will divide 209 and 347 leaving remainder 5 and 7 respectively?
A number is three times another number and their HCF is 8. What is the sum of the squares of the numbers?
The HCF of 2091, 3485 and 4879 is x. The sum of the digits of x is:
If three numbers are in ratio of 3 : 5 : 7 and their LCM is 2415, what is the difference between the second number and the first number?
