Answer :
Sure, let's find [tex]\( f(-3) \)[/tex] and [tex]\( f(4) \)[/tex] for the function [tex]\( f(x) = x^3 - 6x^2 + 2x \)[/tex] using synthetic substitution.
First, for [tex]\( f(-3) \)[/tex]:
1. Write down the coefficients of the polynomial: [tex]\( 1, -6, 2, 0 \)[/tex]. (Here, there is an implied [tex]\( +0 \)[/tex] for the constant term since it's missing.)
2. Use synthetic substitution with [tex]\( x = -3 \)[/tex]:
- Bring down the first coefficient: 1.
- Multiply it by [tex]\(-3\)[/tex] and add it to the next coefficient: [tex]\(-6 + (-3 \times 1) = -6 - 3 = -9\)[/tex].
- Multiply [tex]\(-9\)[/tex] by [tex]\(-3\)[/tex] and add to the next coefficient: [tex]\( 2 + (-3 \times -9) = 2 + 27 = 29\)[/tex].
- Multiply 29 by [tex]\(-3\)[/tex] and add to 0: [tex]\(0 + (-3 \times 29) = 0 - 87 = -87\)[/tex].
The result is [tex]\( f(-3) = -87 \)[/tex].
Next, for [tex]\( f(4) \)[/tex]:
1. Start again with the coefficients: [tex]\( 1, -6, 2, 0 \)[/tex].
2. Use synthetic substitution with [tex]\( x = 4 \)[/tex]:
- Bring down the first coefficient: 1.
- Multiply it by [tex]\(4\)[/tex] and add it to the next coefficient: [tex]\(-6 + (4 \times 1) = -6 + 4 = -2\)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(4\)[/tex] and add it to the next coefficient: [tex]\(2 + (4 \times -2) = 2 - 8 = -6\)[/tex].
- Multiply [tex]\(-6\)[/tex] by [tex]\(4\)[/tex] and add to 0: [tex]\(0 + (4 \times -6) = 0 - 24 = -24\)[/tex].
The result is [tex]\( f(4) = -24 \)[/tex].
So, the values are [tex]\( f(-3) = -87 \)[/tex] and [tex]\( f(4) = -24 \)[/tex]. The correct answer is:
d. [tex]\(-87, -24\)[/tex]
First, for [tex]\( f(-3) \)[/tex]:
1. Write down the coefficients of the polynomial: [tex]\( 1, -6, 2, 0 \)[/tex]. (Here, there is an implied [tex]\( +0 \)[/tex] for the constant term since it's missing.)
2. Use synthetic substitution with [tex]\( x = -3 \)[/tex]:
- Bring down the first coefficient: 1.
- Multiply it by [tex]\(-3\)[/tex] and add it to the next coefficient: [tex]\(-6 + (-3 \times 1) = -6 - 3 = -9\)[/tex].
- Multiply [tex]\(-9\)[/tex] by [tex]\(-3\)[/tex] and add to the next coefficient: [tex]\( 2 + (-3 \times -9) = 2 + 27 = 29\)[/tex].
- Multiply 29 by [tex]\(-3\)[/tex] and add to 0: [tex]\(0 + (-3 \times 29) = 0 - 87 = -87\)[/tex].
The result is [tex]\( f(-3) = -87 \)[/tex].
Next, for [tex]\( f(4) \)[/tex]:
1. Start again with the coefficients: [tex]\( 1, -6, 2, 0 \)[/tex].
2. Use synthetic substitution with [tex]\( x = 4 \)[/tex]:
- Bring down the first coefficient: 1.
- Multiply it by [tex]\(4\)[/tex] and add it to the next coefficient: [tex]\(-6 + (4 \times 1) = -6 + 4 = -2\)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(4\)[/tex] and add it to the next coefficient: [tex]\(2 + (4 \times -2) = 2 - 8 = -6\)[/tex].
- Multiply [tex]\(-6\)[/tex] by [tex]\(4\)[/tex] and add to 0: [tex]\(0 + (4 \times -6) = 0 - 24 = -24\)[/tex].
The result is [tex]\( f(4) = -24 \)[/tex].
So, the values are [tex]\( f(-3) = -87 \)[/tex] and [tex]\( f(4) = -24 \)[/tex]. The correct answer is:
d. [tex]\(-87, -24\)[/tex]
