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  • If 1 and -1 are zeroes of the polynomial ax⁴+bx³+cx²+bx+e prove that a+c+e=b+d

If 1 and -1 are zeroes of the polynomial ax⁴+bx³+cx²+bx+e, prove that a+c+e=b+d

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Given- P(1) = 0 and P(-1) = 0.

  • Substitute x = 1 into the polynomial: P(1) = a(1)4 + b(1)3 + c(1)2 + b(1) + e P(1) = a + b + c + b + e P(1) = a + 2b + c + e. Since P(1) = 0, we get the equation: a + 2b + c + e = 0 (Equation 1)
  • Substitute x = -1 into the polynomial: P(-1) = a(-1)4 + b(-1)3 + c(-1)2 + b(-1) + e P(-1) = a - b + c - b + e P(-1) = a - 2b + c + e. Since P(-1) = 0, we get the equation: a - 2b + c + e = 0 (Equation 2).

Now, subtract Equation 2 from Equation 1: (a + 2b + c + e) - (a - 2b + c + e) = 0

Simplifying: a + 2b + c + e - a + 2b - c - e = 0 → 4b = 0→ b = 0.

Now, substitute b = 0 into Equation 1: a + 2(0) + c + e = 0 → a + c + e = 0.

We have shown that: a + c + e = 0. This is the same as: a + c + e = b + d (since b = 0).

Posted by

Saniya Khatri

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