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Answer :
Sure, let's find [tex]\( f(-3) \)[/tex] and [tex]\( f(4) \)[/tex] using synthetic substitution for the function [tex]\( f(x) = x^4 + x^3 - 3x^2 - x + 12 \)[/tex].
Finding [tex]\( f(-3) \)[/tex]:
1. Write down the coefficients of the polynomial: [tex]\( 1, 1, -3, -1, \)[/tex] and [tex]\( 12 \)[/tex].
2. Use synthetic substitution with [tex]\( x = -3 \)[/tex]:
- Bring down the first coefficient, [tex]\( 1 \)[/tex].
- Multiply by [tex]\(-3\)[/tex] and add to the next coefficient.
- [tex]\( 1 \times -3 = -3 \)[/tex] => next coefficient is [tex]\( 1 + (-3) = -2 \)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(-3\)[/tex] and add to the next coefficient.
- [tex]\(-2 \times -3 = 6 \)[/tex] => next coefficient is [tex]\(-3 + 6 = 3 \)[/tex].
- Multiply [tex]\( 3 \)[/tex] by [tex]\(-3\)[/tex] and add to the next coefficient.
- [tex]\( 3 \times -3 = -9 \)[/tex] => next coefficient is [tex]\(-1 + (-9) = -10 \)[/tex].
- Multiply [tex]\(-10\)[/tex] by [tex]\(-3\)[/tex] and add to the final coefficient.
- [tex]\(-10 \times -3 = 30 \)[/tex] => last term is [tex]\( 12 + 30 = 42 \)[/tex].
The result from synthetic substitution is [tex]\( f(-3) = 42 \)[/tex].
Finding [tex]\( f(4) \)[/tex]:
1. Use the same coefficients: [tex]\( 1, 1, -3, -1, \)[/tex] and [tex]\( 12 \)[/tex].
2. Use synthetic substitution with [tex]\( x = 4 \)[/tex]:
- Bring down the first coefficient, [tex]\( 1 \)[/tex].
- Multiply by [tex]\(4\)[/tex] and add to the next coefficient.
- [tex]\( 1 \times 4 = 4 \)[/tex] => next coefficient is [tex]\( 1 + 4 = 5 \)[/tex].
- Multiply [tex]\( 5 \)[/tex] by [tex]\( 4 \)[/tex] and add to the next coefficient.
- [tex]\( 5 \times 4 = 20 \)[/tex] => next coefficient is [tex]\(-3 + 20 = 17 \)[/tex].
- Multiply [tex]\( 17 \)[/tex] by [tex]\( 4 \)[/tex] and add to the next coefficient.
- [tex]\( 17 \times 4 = 68 \)[/tex] => next coefficient is [tex]\(-1 + 68 = 67 \)[/tex].
- Multiply [tex]\( 67 \)[/tex] by [tex]\(4\)[/tex] and add to the final coefficient.
- [tex]\( 67 \times 4 = 268 \)[/tex] => last term is [tex]\( 12 + 268 = 280 \)[/tex].
The result from synthetic substitution is [tex]\( f(4) = 280 \)[/tex].
Thus, the values of the function are [tex]\( f(-3) = 42 \)[/tex] and [tex]\( f(4) = 280 \)[/tex]. So, the correct answer is (b) 42, 280.
Finding [tex]\( f(-3) \)[/tex]:
1. Write down the coefficients of the polynomial: [tex]\( 1, 1, -3, -1, \)[/tex] and [tex]\( 12 \)[/tex].
2. Use synthetic substitution with [tex]\( x = -3 \)[/tex]:
- Bring down the first coefficient, [tex]\( 1 \)[/tex].
- Multiply by [tex]\(-3\)[/tex] and add to the next coefficient.
- [tex]\( 1 \times -3 = -3 \)[/tex] => next coefficient is [tex]\( 1 + (-3) = -2 \)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(-3\)[/tex] and add to the next coefficient.
- [tex]\(-2 \times -3 = 6 \)[/tex] => next coefficient is [tex]\(-3 + 6 = 3 \)[/tex].
- Multiply [tex]\( 3 \)[/tex] by [tex]\(-3\)[/tex] and add to the next coefficient.
- [tex]\( 3 \times -3 = -9 \)[/tex] => next coefficient is [tex]\(-1 + (-9) = -10 \)[/tex].
- Multiply [tex]\(-10\)[/tex] by [tex]\(-3\)[/tex] and add to the final coefficient.
- [tex]\(-10 \times -3 = 30 \)[/tex] => last term is [tex]\( 12 + 30 = 42 \)[/tex].
The result from synthetic substitution is [tex]\( f(-3) = 42 \)[/tex].
Finding [tex]\( f(4) \)[/tex]:
1. Use the same coefficients: [tex]\( 1, 1, -3, -1, \)[/tex] and [tex]\( 12 \)[/tex].
2. Use synthetic substitution with [tex]\( x = 4 \)[/tex]:
- Bring down the first coefficient, [tex]\( 1 \)[/tex].
- Multiply by [tex]\(4\)[/tex] and add to the next coefficient.
- [tex]\( 1 \times 4 = 4 \)[/tex] => next coefficient is [tex]\( 1 + 4 = 5 \)[/tex].
- Multiply [tex]\( 5 \)[/tex] by [tex]\( 4 \)[/tex] and add to the next coefficient.
- [tex]\( 5 \times 4 = 20 \)[/tex] => next coefficient is [tex]\(-3 + 20 = 17 \)[/tex].
- Multiply [tex]\( 17 \)[/tex] by [tex]\( 4 \)[/tex] and add to the next coefficient.
- [tex]\( 17 \times 4 = 68 \)[/tex] => next coefficient is [tex]\(-1 + 68 = 67 \)[/tex].
- Multiply [tex]\( 67 \)[/tex] by [tex]\(4\)[/tex] and add to the final coefficient.
- [tex]\( 67 \times 4 = 268 \)[/tex] => last term is [tex]\( 12 + 268 = 280 \)[/tex].
The result from synthetic substitution is [tex]\( f(4) = 280 \)[/tex].
Thus, the values of the function are [tex]\( f(-3) = 42 \)[/tex] and [tex]\( f(4) = 280 \)[/tex]. So, the correct answer is (b) 42, 280.
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