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.
Show that 3x + 2 is a factor of 3x2 – x – 2.
Find the value of a, if x – 2 is a factor of 2x5 – 6x4 – 2ax3 + 6ax2 + 4ax + 8.
Using the Factor Theorem, show that (x – 2) is a factor of x3 – 2x2 – 9x + 18. Hence, factorise the expression x3 – 2x2 – 9x + 18 completely.
Using the Factor Theorem, show that (3x + 2) is a factor of 3x3 + 2x2 – 3x – 2. Hence, factorise the expression 3x3 + 2x2 – 3x – 2 completely.
(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.
If (x - 2) is a factor of the expression 2x3 + ax2 + bx - 14 and when the expression is divided by (x - 3), it leaves a remainder 52, find the values of a and b.
In the following problems use the factor theorem to find if g(x) is a factor of p(x):
p(x) = x3 - 3x2 + 4x - 4 and g(x) = x - 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.
