If 2x$^2$ + x - 3 divides x - 2, find the remainder.

Analyze the question: We are dividing the polynomial 2x² + x − 3 by (x − 2) and need the remainder.

Concept (Remainder Theorem): When a polynomial f(x) is divided by (x − a), the remainder is f(a).

Solution:

Here f(x) = 2x² + x − 3 and the divisor is x − 2, so a = 2.

f(2) = 2(2)² + 2 − 3 f(2) = 2(4) + 2 − 3 f(2) = 8 + 2 − 3 f(2) = 7

Check by long division:

• 2x² ÷ x = 2x, and 2x(x − 2) = 2x² − 4x. Subtracting gives 5x − 3.
• 5x ÷ x = 5, and 5(x − 2) = 5x − 10. Subtracting gives −3 − (−10) = 7.

The quotient is 2x + 5 and the remainder is 7.

Answer: The remainder is 7, which is option A.

Tip: When you see "divided by (x − a), find the remainder", don't do long division. Just substitute x = a into the polynomial. If the divisor is (x + a), substitute x = −a. This saves a lot of time in JAMB.