Question

if x4-1 is a factor of ax3+bx2+cx+d show that a+c=0 it's urgent​

Answer 1

Step-by-step explanation :

Cubic polynomial :-

• It is a polynomial of degree 3

• General form, ax³ + bx² + cx + d

• Relationship between zeroes and coefficients :

           Sum of zeroes = -b/a

            Sum of the product of zeroes taken two at a time = c/a

            Product of zeroes = -d/a

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let p(x) = ax³ + bx² + cx + d

(x⁴ - 1) is a factor of p(x)

=> x⁴ - 1 = 0

    x⁴ = 1

    x = ⁴√1

    x = ±1

So, p(1) = 0 , p(-1) = 0

Put x = 1,

a(1)³ + b(1)² + c(1) + d = 0

  a(1) + b(1) + c + d = 0

 a + b + c + d = 0 ---(1)

Put x = -1,

a(-1)³ + b(-1)² + c(-1) + d = 0

 a(-1) + b(1) - c + d = 0

  -a + b - c + d = 0

     b + d = a + c ---(2)

Substitute the value of b+d in equation (1)

    a + b + c + d = 0

    a + c + a + c = 0

     2a + 2c = 0

     2(a + c) = 0

      a + c = 0/2

       a + c = 0

Also, b + d = 0