Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let f(x) = x3 + (a + 1)x2 – (b – 2)x – 6
Since, (x + 1) is a factor of f(x).
∴ Remainder = f(–1) = 0
(–1)3 + (a + 1)(–1)2 – (b – 2)(–1) – 6 = 0
–1 + (a + 1) + (b – 2) – 6 = 0
a + b – 8 = 0 ...(i)
Since, (x – 2) is a factor of f(x).
∴ Remainder = f(2) = 0
(2)3 + (a + 1)(2)2 – (b – 2)(2) – 6 = 0
8 + 4a + 4 – 2b + 4 – 6 = 0
4a – 2b + 10 = 0
2a – b + 5 = 0 ...(ii)
Adding (i) and (ii), we get,
3a – 3 = 0
a = 1
Substituting the value of a in (i), we get,
1 + b – 8 = 0
b = 7
∴ f(x) = x3 + 2x2 – 5x – 6
Now, (x + 1) and (x – 2) are factors of f(x).
Hence, (x + 1)(x – 2) = x2 – x – 2 is a factor of f(x).
x + 3
`x^2 - x - 2")"overline(x^3 + 2x^2 - 5x - 6)`
x3 – x2 – 2x
3x2 – 3x – 6
3x2 – 3x – 6
0
∴ f(x) = x3 + 2x2 – 5x – 6 = (x + 1)(x – 2)(x + 3)
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’.
Show that (x – 1) is a factor of x3 – 7x2 + 14x – 8. Hence, completely factorise the given expression.
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:
x3 + 2ax2 + ax - 1
f 2x3 + ax2 – 11x + b leaves remainder 0 and 42 when divided by (x – 2) and (x – 3) respectively, find the values of a and b. With these values of a and b, factorize the given expression.
Factorize completely using factor theorem:
2x3 – x2 – 13x – 6
While factorizing a given polynomial using the remainder and factor theorem, a student finds that (2x + 1) is a factor of 2x3 + 7x2 + 2x – 3.
- Is the student’s solution correct in stating that (2x + 1) is a factor of the given polynomial?
- Give a valid reason for your answer.
Also, factorize the given polynomial completely.
If (x – a) is a factor of x3 – ax2 + x + 5; the value of a is ______.
For the polynomial x5 – x4 + x3 – 8x2 + 6x + 15, the maximum number of linear factors is ______.
