A quadratic expression in x, when divided by (x – 2) and (x + 3), leaves remainders 13 and 8 respectively. If the expression is divisible by (x – 1), find the quadratic expression.
A quadratic expression in x, when divided by (x – 2) and (x + 3), leaves remainders 13 and 8 respectively. If the expression is divisible by (x – 1), find the quadratic expression. Correct Answer 3x<sup>2</sup> + 4x - 7
Calculation:
Let the quadratic expression be Q(x) = ax2 + bx + c
As given in question the expression is divisible by (x - 1) and leaves remainders 13 and 8 when divided by (x - 2) and (x + 3) respectively.
So, Q(1) = 0 ⇒ a + b + c = 0 ...(1)
Q(2) = 13 ⇒ 4a + 2b + c = 13 ...(2)
Q(-3) = 8 ⇒ 9a - 3b + c = 8 ...(3)
On solving (1), (2), and (3), we get,
a = 3
b = 4
c = -7
∴ Q(x) = 3x2 + 4x -7
Additional Information
Factor theorem:
If R = 0, i.e p(a) = 0, then (x – a) is a factor of p(x) and conversely, if (x – a) is a factor of p(x), then p(a) = 0.
This immediate consequence of the Remainder Theorem is called the Factor Theorem. This can be restated as follows: The number a is a root of p(x) = 0, if and only if (x – a) is a factor of p(x).
