Question

Given (x + 1) and (x – 3) are factors of P(x) = x4 – 5x3 – 7x
2 + cx + d. Determine the values
of the constants c and d.

Answer 1

Question:

Given (x + 1) and (x – 3) are factors of the polynomial p(x) = x⁴ – 5x³ – 7x² + cx + d. Determine the values of the constants c and d.

Answer:

c = 29 , d = 30 .

Note:

• If (x-a) is a factor of the polynomial p(x) ,then x=a is a zero of polynomial p(x) and hence p(a) = 0.

• The possible values of x for which the polynomial p(x) becomes zero are called its zeros.

Solution:

The given polynomial is ;

p(x) = x⁴ – 5x³ – 7x² + cx + d.

Also,

It is given that , (x+1) and (x-3) are the zeros of the given polynomial p(x) .

Since,

(x+1) is a factor of p(x) thus x = -1 is a zero of p(x).

Hence;

=> p(-1) = 0

=> (-1)⁴ - 5•(-1)³ - 7•(-1)² + c•(-1) + d = 0

=> 1 + 5 - 7 - c + d = 0

=> - 1 - c + d = 0

=> d = c + 1 -------(1)

Again,

Since , (x-3) is a factor of p(x) thus x = 3 is a zero of p(x) . Hence ;

=> p(3) = 0

=> 3⁴ - 5•3³ - 7•3² + c•3 + d = 0

=> 81 - 135 - 63 + 3c + d = 0

=> - 117 + 3c + d = 0

=> d = 117 - 3c -------(2)

From eq-(2) and eq-(2) , we get ;

=> c + 1 = 117 - 3c

=> c + 3c = 117 - 1

=> 4c = 116

=> c = 116/4

=> c = 29

Now,

Putting c = 29 in eq-(1) , we get ;

=> d = c + 1

=> d = 29 + 1

=> d = 30

Hence,

The required values of c and d are 29 and 30 respectively .