If-2 and 3 are the zeroes of the quadratic polynomial x + (p + 1)x +q, then find the value of p
Answers
Correct question :
If -2 and 3 are the zeroes of the quadratic polynomial x² + (p + 1)x +q, then find the value of p.
Answer :
p = -2
Step-by-step explanation :
Given polynomial,
p(x) = x² + (p + 1)x + q
- -2 is a zero
Since, -2 is a zero of the polynomial;
substitute x = -2 and the result is zero.
i.e., p(-2) = 0
(-2)² + (p + 1)(-2) + q = 0
4 - 2p - 2 + q = 0
2 - 2p + q = 0
2p - q = 2 --- [ 1 ]
- 3 is a zero
Since, 3 is a zero of the polynomial;
substitute x = 3 and the result is zero.
i.e., p(3) = 0
(3)² + (p + 1)(3) + q = 0
9 + 3p + 3 + q = 0
12 + 3p + q = 0
3p + q = -12 --- [ 2 ]
Add equations [1] and [2]
2p - q + 3p + q = 2 - 12
5p = -10
p = -10/5
p = -2
∴ The value of p is -2.
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[ OR ]
we know,
Relationship between zeroes and coefficients :
⇒ Sum of zeroes = -(x coefficient)/x² coefficient
⇒ Product of zeroes = constant term/x² coefficient
Given polynomial,
x² + (p + 1)x + q
⇒ x² coefficient = 1
x coefficient = (p + 1)
constant term = q
-2 and 3 are zeroes of the polynomial,
Sum of zeroes = -(p + 1)/1
- 2 + 3 = -p - 1
1 = -p - 1
- p = 1 + 1
- p = 2
p = -2
The value of p is -2