Q1. If the two zeroes of the polynomial f(x) = x^3 – 6x^2 + mx + n are 2+ root 5 and 2 - root 5, then find third zero and values of m and n.
Q2. Divide the polynomial f(x) = 5x^2 – 8x^3 + 2 – 15x by the polynomial 2x – 4x^2 + 1 and verify the division algorithm.
PLEASE GIVE ALL STEPS
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Answers
Answer:
x^3-2x+4 will be the answer
Given : f(x) = x³ - 6x² + mx + n 2 + √5 & 2 - √5 are Zeroes
To find : third zero and values of m and n.
Solution :
f(x) = x³ - 6x² + mx + n
2 + √5 & 2 - √5 are Zeroes
let say 3rd root is α
Sum of roots = - (-6)/1 = 6
=> 2 + √5 + 2 - √5 + α = 6
=> α = 2
Products of roots = - n/1
=> (2 + √5) (2 - √5)2 = - n
=> -1 (2) = - n
=> n = 2
putting 2 in f(x)
=> f(2) = 2³ - 6(2)² + 2m + n = 0
=> 2m + n = 16
=> 2m + 2 = 16
=> 2m = 14
=> m = 7
third zero = 2
m = 7 & n = 2
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