x4 + 4×3 – 2×2 + 11x – 6 for x = -3
The Correct Answer and Explanation is:
To evaluate the polynomial expression
x⁴ + 4x³ – 2x² + 11x – 6
for x = -3, substitute -3 in place of x:
Step-by-step Calculation
Expression:
x⁴ + 4x³ – 2x² + 11x – 6
Substitute x = -3:
= (-3)⁴ + 4(-3)³ – 2(-3)² + 11(-3) – 6
1. Evaluate each power and multiplication:
- (-3)⁴ = 81
- (-3)³ = -27
- 4 × (-27) = -108
- (-3)² = 9
- -2 × 9 = -18
- 11 × (-3) = -33
2. Plug in values:
= 81 – 108 – 18 – 33 – 6
3. Perform the operations from left to right:
- 81 – 108 = -27
- -27 – 18 = -45
- -45 – 33 = -78
- -78 – 6 = -84
Final Answer:
-84
Explanation (like in textbooks):
To evaluate a polynomial at a specific value of x, each term is calculated separately by raising the input value to the appropriate power and then multiplying by the coefficient. After all the terms are computed, they are combined through addition or subtraction.
In the polynomial x⁴ + 4x³ – 2x² + 11x – 6, when x = -3, the powers of -3 are calculated first:
- (-3)⁴ = 81 because any negative number raised to an even power becomes positive.
- (-3)³ = -27 because raising a negative number to an odd power results in a negative value.
- (-3)² = 9, again positive because of the even power.
Then, each term is evaluated using multiplication with its coefficient, and finally all terms are summed together.
This step-by-step process ensures accurate computation and helps in understanding the behavior of polynomial functions at specific input values.
