If α, β are the zeroes of the polynomial x2−px+36 and α2+β2 = 9, then what is the value of p?
By BYJU'S Exam Prep
Updated on: October 17th, 2023
If α, β are the zeroes of the polynomial x2−px+36 and α2+β2 = 9, the value of p is 9 or -9. Candidates should formulate polynomial equations in standard form before attempting to solve them. Factor it, then set all of the variable factors to zero once they have all reached zero. The solutions to the original equations are the responses to the derived equations.
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f(x) = x2−px+36 and α2+β2 = 9, Find the value of p.
A polynomial is a mathematical expression that solely uses the operations of addition, subtraction, multiplication, and positive-integer powers of variables. It comprises indeterminates (also known as variables) and coefficients. The question states “If α, β are the zeroes of the polynomial x2−px+36 and α2+β2 = 9, then what is the value of p?”
A polynomial function can only have real roots that are almost equal to its degree. Find a function’s roots by setting the value of the function to zero. Here are the steps for finding the value of p, when α2+β2 = 9:
Given polynomial x2 − px + 36
We know that the standard form of a polynomial is ax2+bx+c
On comparing, we get a = 1, b = -p, c = 36
Given, α and β are the zeroes of the polynomial.
Sum of zeros = α+β = -b/a = p
Product of zeros = αβ = c/a = 36
Now, α2+β2 = (α+β)2 – 2αβ
We know that α2+β2 = 9, substituting the value in the above equation:
9 = p2 – 2*36
9 = p2 – 72
p2 = 81
p = 9, -9
Hence, the value of p is 9 or -9.
Summary:
If α, β are the zeroes of the polynomial x2−px+36 and α2+β2 = 9, then what is the value of p?
The value of p is 9 or -9, if α, β are the zeroes of the polynomial x2−px+36 and α2+β2 = 9. The product of zeros equals c/a and the sum of zeros equals b/a. A quadratic polynomial is represented by the expression k [x2 – (zero-sum)x + (zero product)]. Before attempting to solve them, candidates should write polynomial equations in standard form. Factor it, and when all the variable factors have reached zero, set them all to zero.
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