Advertisements
Advertisements
प्रश्न
Given that x – 2 and x + 1 are factors of f(x) = x3 + 3x2 + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).
Advertisements
उत्तर
f(x)= x3 + 3x2 + ax + b
Since, (x – 2) is a factor of f(x), f(2) = 0
`\implies` (2)3 + 3(2)2 + a(2) + b = 0
`\implies` 8 + 12 + 2a + b = 0
`\implies` 2a + b + 20 = 0 ...(i)
Since, (x + 1) is a factor of f(x), f(–1) = 0
`\implies` (–1)3 + 3(–1)2 + a(–1) + b = 0
`\implies` –1 + 3 – a + b = 0
`\implies` –a + b + 2 = 0 ...(ii)
Subtracting (ii) from (i), we get,
3a + 18 = 0
⇒ a = – 6
Substituting the value of a in (ii), we get,
b = a – 2
= – 6 – 2
= – 8
∴ f(x) = x3 + 3x2 – 6x – 8
Now, for x = –1,
f(x) = f(–1)
= (–1)3 + 3(–1)2 – 6(–1) – 8
= –1 + 3 + 6 – 8
= 0
Hence, (x + 1) is a factor of f(x).
x2 + 2x – 8
`x + 1")"overline(x^3 + 3x^2 - 6x - 8)`
x3 + x2
2x2 – 6x
2x2 + 2x
– 8x – 8
– 8x – 8
0
∴ x3 + 3x2 – 6x – 8 = (x + 1)(x2 + 2x – 8)
= (x + 1)(x2 + 4x – 2x – 8)
= (x + 1)[x(x + 4) – 2(x + 4)]
= (x + 1)(x + 4)(x – 2)
संबंधित प्रश्न
A two digit positive number is such that the product of its digits is 6. If 9 is added to the number, the digits interchange their places. Find the number.
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.
If (x + 1) and (x – 2) are factors of x3 + (a + 1)x2 – (b – 2)x – 6, find the values of a and b. And then, factorise the given expression completely.
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.
Find the values of a and b in the polynomial f(x) = 2x3 + ax2 + bx + 10, if it is exactly divisible by (x+2) and (2x-1).
If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x5 - 3x4 - ax3 + 3ax2 + 2ax + 4.
When 3x2 – 5x + p is divided by (x – 2), the remainder is 3. Find the value of p. Also factorise the polynomial 3x2 – 5x + p – 3.
Prove that (5x + 4) is a factor of 5x3 + 4x2 – 5x – 4. Hence factorize the given polynomial completely.
One factor of x3 – kx2 + 11x – 6 is x – 1. The value of k is ______.
For the polynomial x5 – x4 + x3 – 8x2 + 6x + 15, the maximum number of linear factors is ______.
