Question

The polynomial px³+4x²-3x+q is completely divisible by x²-1;find the values of p and q.also,for these values of p and q factorize the given polynomial completely​

Answer 1

Answer:

The given polynomial is:

px^3 + 4x^2 - 3x + q

It is given that the polynomial is divisibile by (x^2 - 1)

This implies that ,

(x^2 - 1) is the factor of the given polynomial.

To get the zeros of the polynomial, equate (x^2 - 1) to zero.

Thus, we have;

=> x^2 - 1 = 0

=> (x-1)(x+1) = 0

=> x = ± 1

Since, x = ± 1 are the zeros of the given polynomial, thus they will satisfy the given polynomial.

Case:(1) when x = 1

=> p(1)^3 + 4(1)^2 - 3(1) + q = 0

=> p + 4 - 3 + q = 0

=> p + q +1 = 0 -------(1)

Case:(1) when x = -1

=> p(-1)^3 + 4(-1)^2 - 3(-1) + q = 0

=> -p + 4 + 3 + q = 0

=> -p + q +7 = 0 -------(2)

Now,

Adding eq-(1) and (2) , we get;

=> 2q + 8 = 0

=> 2q = - 8

=> q = - 8/2

=> q = - 4

Now ,

Putting q = - 4 in eq-(1), we get;

=> p + q + 1 = 0

=> p - 4 + 1 = 0

=> p - 3 = 0

=> p = 3.

Hence, the required values of p and q are

3 and -4 respectively.

And hence, the given polynomial is:

px^3 + 4x^2 - 3x + q

ie, 3x^3 + 4x^2 - 3x - 4

=> x^2(3x + 4) - (3x + 4)

=> (3x + 4)(x^2 - 1)

=> (3x + 4)(x +1)(x -1)

Hence, the factorised form of the polynomial is:

(3x + 4)(x +1)(x -1)