if alpha and beta are the zeroes of the quadratic polynomial f(t)=t2-4t+3, find the value of alpha4*beta3+alpha3*beta4
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Appropriate Question :
If α and β are the zeroes of the quadratic polynomial f(t) = t² - 4t + 3, find the value of α⁴β³ + α³β⁴
Answer :
The required value of α⁴β³ + α³β⁴ is 108.
Step-by-step explanation :
Given :
α and β are the zeroes of the quadratic polynomial f(t) = t² - 4t + 3
To find :
the value of α⁴β³ + α³β⁴
Solution :
Let's find the values of α and β first.
Finding the zeroes of the polynomial f(t) = t² - 4t + 3 :
By factorization,
t² - 4t + 3 = 0
t² - t - 3t + 3 = 0
t(t - 1) - 3(t - 1) = 0
(t - 1) (t - 3) = 0
⇒ t - 1 = 0 ; t = 1
⇒ t - 3 = 0 ; t = 3
∴ 1 and 3 are the zeroes of the given polynomial f(t) = t² - 4t + 3.
So, let α = 1 and β = 3,
And substitute the values,
α⁴β³ + α³β⁴
[ (1)⁴ × (3)³ ] + [ (1)³ × (3)⁴ ]
[ 1 × 27 ] + [ 1 × 81 ]
27 + 81
108
The required value of α⁴β³ + α³β⁴ is 108.
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