Advertisements
Advertisements
Question
Find the value of the constant a and b, if (x – 2) and (x + 3) are both factors of expression x3 + ax2 + bx - 12.
Advertisements
Solution
Expression x3 + ax2 + bx - 12
(x - 2) is a factor i.e, at x = 2
the remainder will be xero
⇒ (2)3 + a(2)2 + b(2) - 12 = 0
⇒ 8 + 4a + 2b - 12 = 0
⇒ 4a + 2b = 4
⇒ 2a + b = 2 ...(i)
when x + 3 is a factor i.e., at x = -3 the remainder will be zero.
⇒ (-3)3 + a(-3)2 + b(-3) -12 = 0
⇒ -27 + 9a - 3b - 12 = 0
⇒ 9a - 3b = 39
⇒ 3a - b = 13 ...(ii)
Solving (i) and (ii) simultaneously
2a + b = 2
By adding
3a - b = 13
5a = 15
a = 3
Substituting the value of a in the equation (i)
⇒ 2 x 3 + b = 2
⇒ 6 + b = 2
⇒ b = 2 - 6 = -4
⇒ a = 3, b = -4.
RELATED QUESTIONS
Show that x – 2 is a factor of 5x2 + 15x – 50.
Find the value of ‘a’, if (x – a) is a factor of x3 – ax2 + x + 2.
Prove by factor theorem that
(x-2) is a factor of 2x3- 7x -2
In the following problems use the factor theorem to find if g(x) is a factor of p(x):
p(x) = x3 + x2 + 3x + 175 and g(x) = x + 5.
Show that 2x + 7 is a factor of 2x3 + 5x2 - 11 x - 14. Hence factorise the given expression completely, using the factor theorem.
Using factor theorem, show that (x - 3) is a factor of x3 - 7x2 + 15x - 9, Hence, factorise the given expression completely.
Determine whether (x – 1) is a factor of the following polynomials:
x3 + 5x2 – 10x + 4
Find the value of 'a' if x – a is a factor of the polynomial 3x3 + x2 – ax – 81.
Factors of 3x3 – 2x2 – 8x are ______.
Is (x – 2) a factor of x3 – 4x2 – 11x + 30?
