Advertisements
Advertisements
Question
(3x + 5) is a factor of the polynomial (a – 1)x3 + (a + 1)x2 – (2a + 1)x – 15. Find the value of ‘a’, factorise the given polynomial completely.
Advertisements
Solution
Let f(x) = (a – 1)x3 + (a + 1)x2 – (2a + 1)x – 15
It is given that (3x + 5) is a factor of f(x).
∴ Remainder = 0
`f((-5)/3) = 0`
`(a - 1)(-5/3)^3 + (a + 1)((-5)/3)^2 - (2a + 1)((-5)/3) - 15 = 0`
`(a - 1)((-125)/27) + (a + 1)(a + 1)(25/9) - (2a + 1)((-5)/3) - 15 = 0`
`(-125(a-1) + 75(a+1) + 45(2a+1) - 405)/27 = 0`
–125a + 125 + 75a + 75 + 90a + 45 – 405 = 0
40a – 160 = 0
40a = 160
a = 4
∴ f(x) = (a – 1)x3 + (a + 1)x2 – (2a + 1)x – 15
= 3x3 + 5x2 – 9x – 15
x2 – 3
`3x + 5")"overline(3x^3 + 5x^2 - 9x - 15)`
3x3 + 5x2
– 9x – 15
– 9x – 15
0
∴ 3x3 + 5x2 – 9x – 15 = (3x + 5)(x2 – 3)
= `(3x + 5)(x + sqrt(3))(x - sqrt(3))`
RELATED QUESTIONS
Show that x – 2 is a factor of 5x2 + 15x – 50.
If 2x + 1 is a factor of 2x2 + ax – 3, find the value of a.
What should be subtracted from 3x3 – 8x2 + 4x – 3, so that the resulting expression has x + 2 as a factor?
Find the value of m ·when x3 + 3x2 -m x +4 is exactly divisible by (x-2)
In the following problems use the factor theorem to find if g(x) is a factor of p(x):
p(x) = 2x3 + 4x + 6 and g(x) = x + 1
Show that (x – 1) is a factor of x3 – 5x2 – x + 5 Hence factorise x3 – 5x2 – x + 5.
Show that (x – 3) is a factor of x3 – 7x2 + 15x – 9. Hence factorise x3 – 7x2 + 15 x – 9
Show that 2x + 7 is a factor of 2x3 + 5x2 – 11x – 14. Hence factorise the given expression completely, using the factor theorem.
Check if (x + 2) and (x – 4) are the sides of a rectangle whose area is x2 – 2x – 8 by using factor theorem
If x – 3 is a factor of x2 + kx + 15; the value of k is ______.
