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Answers
Answer:
Step-by-step explanation:
Given :
The two zeroes of the polynomial p(x) = Lx⁴ + Mx³ + Nx² + Rx + P are, 1 & -1, respectively,.
To Prove :
L + N + P = M + R = 0
Proof :
p(x) = Lx⁴ + Mx³ + Nx² + Rx + P
The zeroes are 1 & -1,.
It means,
By substituting x = 1 & -1 ,.
We'll p(x) = 0
So,
By substituting x = 1 ,
We get,
p(x) = Lx⁴ + Mx³ + Nx² + Rx + P
⇒ p(1) = L(1)⁴ + M(1)³ + N(1)² + R(1) + P
⇒ 0 = L(1) + M(1) + N(1) + R(1) + P
⇒ 0 = L + M + N + R + P ...(i)
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By substituting x = -1 ,
We get,
p(x) = Lx⁴ + Mx³ + Nx² + Rx + P
p(-1) = L(-1)⁴ + M(-1)³ + N(-1)² + R(-1) + P
⇒ 0 = L(1) + M(-1) + N(1) + R(-1) + P
⇒ 0 = L - M + N - R + P ..(ii)
⇒ M + R = L + N + P ...(iii)
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By comparing (i) & (ii),
⇒ L + M + N + R + P = L - M + N - R + P
⇒ L + M + N + R + P - (L - M + N - R + P) = 0
⇒ L - L + M + M + N - N + R + R + P - P = 0
⇒ 2M + 2R = 0
⇒ 2(M + R) = 0 ⇒ M + R = 0 ...(iv)
By (iii) & (iv),
We get,
L + N + P = M + R = 0
Hence, Proved,.