If the polynomial 3x^2 + ax + 4 is exactly divisible by x − 5, meaning that x − 5 is a factor of the polynomial, what is the value of the constant a?

Introduction / Context:
This algebra question involves the factor theorem for polynomials. If x − 5 is a factor of the quadratic polynomial 3x^2 + ax + 4, then substituting x = 5 into the polynomial must give zero. This condition allows us to find the unknown coefficient a.

Given Data / Assumptions:

• Polynomial: P(x) = 3x^2 + ax + 4.
• x − 5 is a factor of P(x).
• Therefore, P(5) = 0 by the factor theorem.

Concept / Approach:
The factor theorem states that if x − r is a factor of a polynomial P(x), then P(r) = 0. Here, r is 5. We substitute x = 5 into the polynomial and set the result equal to zero, then solve the resulting linear equation for a. This method avoids long division and directly uses the property of polynomial factors.

Step-by-Step Solution:
Step 1: Write down the condition from the factor theorem: P(5) = 0. Step 2: Compute P(5) using P(x) = 3x^2 + ax + 4. Step 3: Substitute x = 5: P(5) = 3 * (5^2) + a * 5 + 4. Step 4: Calculate 5^2 = 25, so 3 * 25 = 75. Step 5: The expression becomes P(5) = 75 + 5a + 4. Step 6: Simplify: P(5) = 79 + 5a. Step 7: Set P(5) equal to zero because x − 5 is a factor: 79 + 5a = 0. Step 8: Solve for a: 5a = −79, so a = −79 / 5. Step 9: Convert the fraction to decimal: −79 / 5 = −15.8. Step 10: Therefore, the required value of a is −15.8.

Verification / Alternative check:
We can quickly verify by substituting a = −15.8 back into the polynomial and evaluating at x = 5. P(5) = 3 * 25 + (−15.8) * 5 + 4 = 75 − 79 + 4 = 0. Since P(5) is zero, x − 5 is indeed a factor, confirming that a = −15.8 is correct.

Why Other Options Are Wrong:
Values like −5, −12, −15.6 and −20 do not satisfy the condition that P(5) = 0. Substituting any of these into P(x) and evaluating at x = 5 yields a nonzero result, meaning x − 5 would not be a factor of the polynomial.

Common Pitfalls:
Students sometimes forget to use the factor theorem and attempt polynomial long division, which is more time consuming and can lead to arithmetic mistakes. Others may incorrectly simplify the expression 3 * 25 + 4 or mis-handle the sign when solving 79 + 5a = 0. Always write the intermediate step clearly as 79 + 5a and solve carefully for a.

Final Answer:
The value of the constant is a = −15.8.