Answer :
To find the values of [tex]\( f(-5) \)[/tex] and [tex]\( f(2) \)[/tex] for the function [tex]\( f(x) = 3x^4 + x^3 - 2x^2 + x + 12 \)[/tex], we can use synthetic substitution, which is a method used to evaluate a polynomial at a given value of [tex]\( x \)[/tex].
Step-by-step solution for [tex]\( f(-5) \)[/tex]:
1. Set up the coefficients of the polynomial. The polynomial [tex]\( f(x) = 3x^4 + x^3 - 2x^2 + x + 12 \)[/tex] has coefficients: 3, 1, -2, 1, 12.
2. Synthetic substitution for [tex]\( x = -5 \)[/tex]:
- Start with the first coefficient: 3.
- Multiply it by [tex]\(-5\)[/tex] and add the next coefficient:
[tex]\[
(3) \times (-5) + 1 = -15 + 1 = -14
\][/tex]
- Multiply [tex]\(-14\)[/tex] by [tex]\(-5\)[/tex] and add the next coefficient:
[tex]\[
(-14) \times (-5) + (-2) = 70 - 2 = 68
\][/tex]
- Multiply 68 by [tex]\(-5\)[/tex] and add the next coefficient:
[tex]\[
(68) \times (-5) + 1 = -340 + 1 = -339
\][/tex]
- Multiply [tex]\(-339\)[/tex] by [tex]\(-5\)[/tex] and add the last coefficient:
[tex]\[
(-339) \times (-5) + 12 = 1695 + 12 = 1707
\][/tex]
So, [tex]\( f(-5) = 1707 \)[/tex].
Step-by-step solution for [tex]\( f(2) \)[/tex]:
1. Set up the coefficients of the polynomial. The polynomial [tex]\( f(x) = 3x^4 + x^3 - 2x^2 + x + 12 \)[/tex] has coefficients: 3, 1, -2, 1, 12.
2. Synthetic substitution for [tex]\( x = 2 \)[/tex]:
- Start with the first coefficient: 3.
- Multiply it by 2 and add the next coefficient:
[tex]\[
(3) \times 2 + 1 = 6 + 1 = 7
\][/tex]
- Multiply 7 by 2 and add the next coefficient:
[tex]\[
(7) \times 2 + (-2) = 14 - 2 = 12
\][/tex]
- Multiply 12 by 2 and add the next coefficient:
[tex]\[
(12) \times 2 + 1 = 24 + 1 = 25
\][/tex]
- Multiply 25 by 2 and add the last coefficient:
[tex]\[
(25) \times 2 + 12 = 50 + 12 = 62
\][/tex]
So, [tex]\( f(2) = 62 \)[/tex].
Therefore, the evaluated values are [tex]\( f(-5) = 1707 \)[/tex] and [tex]\( f(2) = 62 \)[/tex].
Step-by-step solution for [tex]\( f(-5) \)[/tex]:
1. Set up the coefficients of the polynomial. The polynomial [tex]\( f(x) = 3x^4 + x^3 - 2x^2 + x + 12 \)[/tex] has coefficients: 3, 1, -2, 1, 12.
2. Synthetic substitution for [tex]\( x = -5 \)[/tex]:
- Start with the first coefficient: 3.
- Multiply it by [tex]\(-5\)[/tex] and add the next coefficient:
[tex]\[
(3) \times (-5) + 1 = -15 + 1 = -14
\][/tex]
- Multiply [tex]\(-14\)[/tex] by [tex]\(-5\)[/tex] and add the next coefficient:
[tex]\[
(-14) \times (-5) + (-2) = 70 - 2 = 68
\][/tex]
- Multiply 68 by [tex]\(-5\)[/tex] and add the next coefficient:
[tex]\[
(68) \times (-5) + 1 = -340 + 1 = -339
\][/tex]
- Multiply [tex]\(-339\)[/tex] by [tex]\(-5\)[/tex] and add the last coefficient:
[tex]\[
(-339) \times (-5) + 12 = 1695 + 12 = 1707
\][/tex]
So, [tex]\( f(-5) = 1707 \)[/tex].
Step-by-step solution for [tex]\( f(2) \)[/tex]:
1. Set up the coefficients of the polynomial. The polynomial [tex]\( f(x) = 3x^4 + x^3 - 2x^2 + x + 12 \)[/tex] has coefficients: 3, 1, -2, 1, 12.
2. Synthetic substitution for [tex]\( x = 2 \)[/tex]:
- Start with the first coefficient: 3.
- Multiply it by 2 and add the next coefficient:
[tex]\[
(3) \times 2 + 1 = 6 + 1 = 7
\][/tex]
- Multiply 7 by 2 and add the next coefficient:
[tex]\[
(7) \times 2 + (-2) = 14 - 2 = 12
\][/tex]
- Multiply 12 by 2 and add the next coefficient:
[tex]\[
(12) \times 2 + 1 = 24 + 1 = 25
\][/tex]
- Multiply 25 by 2 and add the last coefficient:
[tex]\[
(25) \times 2 + 12 = 50 + 12 = 62
\][/tex]
So, [tex]\( f(2) = 62 \)[/tex].
Therefore, the evaluated values are [tex]\( f(-5) = 1707 \)[/tex] and [tex]\( f(2) = 62 \)[/tex].
