If $f(x) = 3x^3 + 4x^2 + x - 8$, what is the value of f(-2)?

Analysis

To find the value of $f(-2)$, we need to perform a process called substitution. This means we replace every instance of the variable $x$ in the given polynomial with the value $-2$ and then simplify the expression following the BODMAS/PEMDAS rule.

Concept: Polynomial Evaluation

A polynomial $f(x)$ is evaluated at a specific point $a$ by calculating $f(a)$. When dealing with negative numbers raised to powers, remember:

• A negative number raised to an odd power remains negative (e.g., $(-2)^3 = -8$).
• A negative number raised to an even power becomes positive (e.g., $(-2)^2 = 4$).

Step-by-Step Solution

Given: $f(x) = 3x^3 + 4x^2 + x - 8$

Step 1: Substitute $x = -2$ into the equation $f(-2) = 3(-2)^3 + 4(-2)^2 + (-2) - 8$

Step 2: Evaluate the powers

• $(-2)^3 = -2 \times -2 \times -2 = -8$
• $(-2)^2 = -2 \times -2 = 4$

Now substitute these back: $f(-2) = 3(-8) + 4(4) - 2 - 8$

Step 3: Perform the multiplications

• $3 \times -8 = -24$
• $4 \times 4 = 16$

Now the expression is: $f(-2) = -24 + 16 - 2 - 8$

Step 4: Simplify the addition and subtraction

• $-24 + 16 = -8$
• $-8 - 2 = -10$
• $-10 - 8 = -18$

The value of $f(-2)$ is -18.

Conclusion

Comparing our result to the given options: A. -24 B. 30 C. -18 D. -50

The correct option is C.

Exam Tip

When substituting negative numbers into polynomials, always use brackets. This prevents common sign errors, especially when dealing with terms like $-x$ or negative coefficients. Also, double-check your arithmetic for the final addition/subtraction, as that is where most students lose marks in JAMB!