Doubt in maths - polynomials?

Question

how do i solve these??
1. if the polynomial 6x^4 + 8x^3 - 5x^2 + ax + b is exactly divisible by 2x^2 - 5, then find the values of a and b.
2. find the values of p and q such that (x-1) and (x+2) are the factors of x^3 + 10x^2 + px + q
3. if 49x^2 - p = (7x - 1/3) (7x+1/3) find the value of p
4. if alpha, beta and gamma are the roots of a polynomial 6x^3 + 3x^2 - 5x +1 find the value of alpha^-1 + beta ^ -1 + gamma ^-1.

Answer 1

1)

The polynomial is .

Since divides the polynomial, you can find the roots of the polynomial by equating to zero.

.

The roots of the polynomial are .

So, we have .

Let us take .

Therefore, .

.

Let us say it is equation 1.

Now  .

Let us say it is equation 2.

Continues....

Answer 2

Solve equation 1 and 2 for the values of a, b.

Add equation 1 and 2.

Now to find the value of a substitute in equation 1.

.

The values of a and b are , .

Comments

The values of a and b are , ----> a = - 20 and b = - 25.

 

Answer 3

2)

The polynomial is .

The given factors of the polynomial are .

Since factors of a polynomial also divides the polynomial.

So, equate the factors to zero and find the roots of the polynomial.

.

The roots of the polynomial are .

Now we have .

Let

Let

Solve equation (1) and (2) for the values of p, q.

Subtract equation (2) from (1).

Substitute in equation (1) and find q.

So, the value of p  and q  are respectively.

Answer 4

3)

The expression is .

The value of  p is .

Answer 5

The polynomial is .

are the roots of the polynomial.

Therefore, .

.

.

Similarly we have .

Now find .

.

                        .

Therefore, .