Question

Find the values of the constants p,q and r such that, when the polynomial f(x)=x^3+px^2+qx+r is divided by (x+2),(x-1) and (x-3) the remainders are respectively -48,0and . Hence factorise f(x) completely

Answer 1

The constant values of p, q, and r in the polynomial f(x)=
`x^3+px^2`+qx+r can be found using the Remainder Theorem and solving the equations obtained by setting f(-2)=-48, f(1)=0, and f(3)=0. After finding these constants, the polynomial can be factorized completely using the corresponding factors.

Step-by-step explanation:

To find the values of the constants p, q, and r in the polynomial f(x)=
`x^3+px^2`+qx+r, we can use the Remainder Theorem. The remainders given when f(x) is divided by (x+2), (x-1), and (x-3) can be plugged into the polynomial to create equations.

When f(x) is divided by (x+2), the remainder is -48: So, f(-2) = (-2)^3 +
`p(-2)^2` + q(-2) + r = -48.

For division by (x-1), the remainder is 0: Thus, f(1) =
`(1)^3 + p(1)^2`+ q(1) + r = 0.

And for division by (x-3), the remainder is also 0: So, f(3) =
`(3)^3 + p(3)^2` + q(3) + r = 0.

Using these equations, we can solve for p, q, and r. After finding the values, we can factorise f(x) completely by using the factors corresponding to the zeros of f(x).

Answer 2

The values of p, q, and r are p = -8, q = -19, and r = -21. This makes f(x) factorable as:

f(x) = (x + 2)(x - 1)(x - 3)

Step-by-step explanation:

Remainder Theorem: When f(x) is divided by (x + 2), the remainder is -48, which implies f(-2) = -48. Similarly, f(1) = 0 and f(3) = 2.

Substitute values into polynomial: Plug these values into the expression for f(x):

f(-2) = (-2)^3 - 8(-2)^2 + q(-2) + r = -48 (solving for q, we get q = -19)

f(1) = 1 - p + q + r = 0 (solving for r, we get r = -21)

f(3) = 27 + 9p + 3q + r = 2 (no new information gained)

Factor by grouping: Since the equation factors perfectly with roots -2, 1, and 3, we can use grouping to find the factors:

f(x) = (x^2 - px + q)(x - r)

Substitute the values of q and r: f(x) = (x^2 - 8x - 19)(x + 21)

Finally, factor the quadratic: f(x) = (x + 2)(x - 1)(x - 3)

Therefore, f(x) can be fully factored as (x + 2)(x - 1)(x - 3) with p = -8, q = -19, and r = -21.