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Answer:
Therefore the solution is = k{-7,5,1} where k ∈R
Step-by-step explanation:
Given that,
f₁(x) =x
f₂(x)= x²
f₃(x)= 7x - 5x²
Also,
g(x) = c₁f₁(x)+c₂f₂(x)+c₃f₃(x)
Putting the values of f₁(x), f₂(x) and f₃(x).
g(x) = c₁.x+c₂x²+c₃(7x-5x²)
Given condition that g(x)= 0
∴ c₁.x+c₂x²+c₃(7x-5x²)=0
⇒(c₁+7c₃)x +(c₂-5c₃)x² = 0
Comparing the coefficients of x and x²
∴c₁+7c₃=0 and c₂-5c₃ =0
Let c₃= k [k∈R]
Then c₁ = -7k and c₂=5k
Therefore the solution is = { c₁,c₂,c₃}
= {-7k, 5k, k}
=k{-7,5,1}
do a FOIL on it to get the product, or just use the binomial theorem to expand the binomial and simplify