Advertisements
Advertisements
Question
Factorise the expression f(x) = 2x3 – 7x2 – 3x + 18. Hence, find all possible values of x for which f(x) = 0.
Advertisements
Solution
f(x) = 2x3 – 7x2 – 3x + 18
For x = 2,
f(x) = f(2)
= 2(2)3 – 7(2)2 – 3(2) + 18
= 16 – 28 – 6 + 18
= 0
Hence, (x – 2) is a factor of f(x).
2x2 – 3x – 9
`x - 2")"overline(2x^3 - 7x^2 - 3x + 18)`
2x3 – 4x2
– +
– 3x2 – 3x
– 3x2 + 6x
+ –
– 9x + 18
– 9x + 18
+ –
0
∴ 2x3 – 7x2 – 3x + 18 = (x – 2)(2x2 – 3x – 9)
= (x – 2)(2x2 – 6x + 3x – 9)
= (x – 2)[2x(x – 3) + 3(x – 3)]
= (x – 2)(x – 3)(2x + 3)
Now, f(x) = 0
⇒ 2x3 – 7x2 – 3x + 18 = 0
⇒ (x – 2)(x – 3)(2x + 3) = 0
⇒ `x = 2, 3, (-3)/2`
RELATED QUESTIONS
Using the Remainder Theorem, factorise each of the following completely.
3x3 + 2x2 – 23x – 30
Show that (x – 1) is a factor of x3 – 7x2 + 14x – 8. Hence, completely factorise the given expression.
Find the value of a and b so that the polynomial x3 - ax2 - 13x + b has (x - 1) (x + 3) as factor.
In the following two polynomials. Find the value of ‘a’ if x + a is a factor of each of the two:
x3 + ax2 − 2x + a + 4
In the following two polynomials, find the value of ‘a’ if x – a is a factor of each of the two:
x6 - ax5 + x4 - ax3 + 3a + 2
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 – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x3 + 2ax2 + ax - 1
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.
For the polynomial x5 – x4 + x3 – 8x2 + 6x + 15, the maximum number of linear factors is ______.
If f(x) = 3x + 8; the value of f(x) + f(– x) is ______.
