If α and β are the zeroes of p(x) = 5x2 – 6x – 2, find a quadratic polynomial whose zeroes are 5α + 2 and 5β + 2.
Answers
Given : α and β are zeroes of polynomial p(x) = 5x² - 6x - 2.
To find : a quadratic polynomial whose zeroes are 5α + 2 and 5β + 2
solution : first find sum and product of zeroes of quadratic polynomial, p(x) = 5x² - 6x - 2
sum of zeroes = α + β = -(-6)/5 = 6/5
product of zeroes = αβ = -2/5
zeroes of new quadratic polynomial are (5α + 2) and (5β + 2)
so sum of zeroes = (5α + 2) + (5β + 2)
= 5(α + β) + 4
= 5 × 6/5 + 4
= 10
product of zeroes = (5α + 2)(5β + 2)
= 25αβ + 10α + 10β + 4
= 25 × -2/5 + 10(α + β) + 4
= -10 + 10 × 6/5 + 4
= -10 + 12 + 4
= 6
quadratic polynomial is x² - (sum of zeroes)x + product of zeroes
= x² - 10x + 6
Therefore the required quadratic polynomial is x² - 10x + 6 .
Answer:
Step-by-step explanation:
Given : α and β are zeroes of polynomial p(x) = 5x² - 6x - 2.
To find : a quadratic polynomial whose zeroes are 5α + 2 and 5β + 2
solution : first find sum and product of zeroes of quadratic polynomial, p(x) = 5x² - 6x - 2
sum of zeroes = α + β = -(-6)/5 = 6/5
product of zeroes = αβ = -2/5
zeroes of new quadratic polynomial are (5α + 2) and (5β + 2)
so sum of zeroes = (5α + 2) + (5β + 2)
= 5(α + β) + 4
= 5 × 6/5 + 4
= 10
product of zeroes = (5α + 2)(5β + 2)
= 25αβ + 10α + 10β + 4
= 25 × -2/5 + 10(α + β) + 4
= -10 + 10 × 6/5 + 4
= -10 + 12 + 4
= 6
quadratic polynomial is x² - (sum of zeroes)x + product of zeroes
= x² - 10x + 6
Therefore the required quadratic polynomial is x² - 10x + 6 .