Find the polynomial if given q(x) = x$^2$ - x - 5, d(x) = 3x - 1 and r(x) = 7.

1. Analyze the question

You are given the quotient $q(x)$, the divisor $d(x)$, and the remainder $r(x)$ of a polynomial division. You need to find the original polynomial, which we can call $P(x)$.

2. Explain the concept

In polynomial division, the relationship between the dividend $P(x)$, the divisor $d(x)$, the quotient $q(x)$, and the remainder $r(x)$ is defined by the Division Algorithm:

To find the polynomial, we simply need to multiply the divisor by the quotient and then add the remainder.

3. Show the solution

Given:

• $q(x) = x^2 - x - 5$
• $d(x) = 3x - 1$
• $r(x) = 7$

Step 1: Multiply $d(x)$ and $q(x)$ $P(x) = (3x - 1)(x^2 - x - 5) + 7$

Expand the brackets: $= 3x(x^2 - x - 5) - 1(x^2 - x - 5) + 7$ $= (3x^3 - 3x^2 - 15x) - (x^2 - x - 5) + 7$

Step 2: Simplify the expression $= 3x^3 - 3x^2 - 15x - x^2 + x + 5 + 7$

Group the like terms: $= 3x^3 + (-3x^2 - x^2) + (-15x + x) + (5 + 7)$ $= 3x^3 - 4x^2 - 14x + 12$

4. Identify the answer

Comparing our result with the given options, the correct polynomial is $3x^3 - 4x^2 - 14x + 12$. This matches Option A.

5. Tip

When multiplying polynomials, always be careful with the negative signs. A common mistake in JAMB exams is forgetting to distribute the negative sign to every term inside the second bracket (e.g., $-1 \times -5$ becomes $+5$). Always double-check your signs before adding the remainder!