Find the remainder when 2x3 - 11x2 + 8x - 1 is divided by x + 3

1. Analyze the Question

We are asked to find the remainder when the polynomial $P(x) = 2x^3 - 11x^2 + 8x - 1$ is divided by the linear factor $(x + 3)$.

2. Explain the Concept: The Remainder Theorem

The Remainder Theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, the remainder is simply $P(a)$.

In this case, we are dividing by $(x + 3)$. To find the value of $a$, we set the divisor to zero: $x + 3 = 0$ $x = -3$

Therefore, the remainder is equal to $P(-3)$.

3. Show the Solution

Substitute $x = -3$ into the polynomial $P(x) = 2x^3 - 11x^2 + 8x - 1$:

Step 1: Substitute -3 for x $P(-3) = 2(-3)^3 - 11(-3)^2 + 8(-3) - 1$

Step 2: Evaluate the powers $(-3)^3 = -27$ $(-3)^2 = 9$

Step 3: Multiply the terms $P(-3) = 2(-27) - 11(9) + 8(-3) - 1$ $P(-3) = -54 - 99 - 24 - 1$

Step 4: Combine the values $-54 - 99 = -153$ $-153 - 24 = -177$ $-177 - 1 = -178$

The remainder is -178.

4. Identify the Answer

Comparing our result to the given options: A. -871 B. -781 C. -187 D. -178

The correct option is D.

5. Exam Tip

When using the Remainder Theorem, be extremely careful with negative signs! Students often make mistakes when squaring or cubing negative numbers. Remember:

• A negative number raised to an even power becomes positive.
• A negative number raised to an odd power remains negative.
• Always write out the substitution clearly in brackets to avoid sign errors.