Advertisements
Advertisements
Question
The polynomial 3x3 + 8x2 – 15x + k has (x – 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.
Advertisements
Solution
Given, P(x) = 3x3 + 8x2 – 15x + k
Put x – 1 = 0
x = 1
Now, P(1) = 3(1)3 + 8(1)2 – 15(1) + k = 0
⇒ 3 + 8 – 15 + k = 0
⇒ – 4 + k = 0
⇒ k = 4
Hence, k = 4
P(x) = 3x3 + 8x2 – 15x + 4
`x - 1")"overline(3x^3 + 8x^2 - 15x + 4)(3x^2 + 11x - 4`
3x3 – 3x2
– +
11x2 – 15x
11x2 – 11x
– +
– 4x + 4
– 4x + 4
+ –
x
∴ 3x3 + 8x2 – 15x + 4 = (x – 1) (3x2 + 11x – 4)
= (x – 1) (3x2 + 12x – x – 4)
= (x – 1) [3x(x + 4) – 1(x + 4)]
= (x – 1) (3x – 1) (x + 4)
RELATED QUESTIONS
When divided by x – 3 the polynomials x3 – px2 + x + 6 and 2x3 – x2 – (p + 3) x – 6 leave the same remainder. Find the value of ‘p’.
Using the Remainder Theorem, factorise each of the following completely.
3x3 + 2x2 – 23x – 30
Find the number that must be subtracted from the polynomial 3y3 + y2 – 22y + 15, so that the resulting polynomial is completely divisible by y + 3.
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.
If the polynomials ax3 + 4x2 + 3x - 4 and x3 - 4x + a leave the same remainder when divided by (x - 3), find the value of a.
If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x2 - 3x + 5a
If (x + 3) and (x – 4) are factors of x3 + ax2 – bx + 24, find the values of a and b: With these values of a and b, factorise the given expression.
One factor of x3 – kx2 + 11x – 6 is x – 1. The value of k is ______.
If (x – a) is a factor of x3 – ax2 + x + 5; the value of a is ______.
If f(x) = 3x + 8; the value of f(x) + f(– x) is ______.
