If α and β are Zeros of Polynomial x²+6x+9, then Form a Quadratic Polynomial whose Zeros are -alpha, -beta.
By BYJU'S Exam Prep
Updated on: October 17th, 2023
If α and β are zeroes of polynomial x²+6x+9, then form a quadratic polynomial whose zeroes are -α,-β.
Here are the steps to form a quadratic polynomial whose zeroes are -α and -β, given that α and β are zeroes of the polynomial x² + 6x + 9:
- Identify the factors of the polynomial x² + 6x + 9, which are (x + α) and (x + β).
- Change the signs of the zeroes to obtain the factors for the new polynomial: (x – (-α)) and (x – (-β)).
- Simplify the expressions: (x + α) and (x + β).
- Multiply the factors to obtain the quadratic polynomial: P(x) = (x + α)(x + β).
- Expand the product: P(x) = x² + (α + β)x + αβ.
- Substitute the values of α and β.
- Simplify the expression to obtain the final quadratic polynomial.
Table of content
If α and β are Zeros of Polynomial x²+6x+9, then Form a Quadratic Polynomial whose Zeros are -alpha, -beta
If α and β are the zeroes of the quadratic polynomial x² + 6x + 9, then we can form a quadratic polynomial whose zeroes are -α and -β.
P(x) has zeroes α and β, then (x – α) and (x – β) are factors of P(x).
Given that the zeroes of the polynomial x² + 6x + 9 are α and β, we have the factors (x – α) and (x – β).
To find a quadratic polynomial with zeroes -α and -β, we can simply change the signs of the zeroes. Therefore, the factors for the new polynomial will be (x + α) and (x + β).
To find the quadratic polynomial, we multiply these factors:
P(x) = (x + α)(x + β)
Expanding this product:
P(x) = x² + (α + β)x + αβ
Since α and β are the zeroes of x² + 6x + 9,
α + β = -6, and
αβ = 9
Substituting these values:
P(x) = x² – 6x + 9
Therefore, the quadratic polynomial with zeroes -α and -β is x² – 6x + 9.
Answer:
If α and β are Zeros of Polynomial x²+6x+9, then a Quadratic Polynomial whose Zeros are -alpha, -beta is x² – 6x + 9
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