Answer :
Let's use synthetic substitution to find [tex]\( f(-3) \)[/tex] and [tex]\( f(4) \)[/tex] for the function [tex]\( f(x) = 3x^4 - 4x^3 + 3x^2 - 5x - 3 \)[/tex].
### Synthetic Substitution for [tex]\( f(-3) \)[/tex]:
1. Write down the coefficients:
- The polynomial is [tex]\( 3x^4 - 4x^3 + 3x^2 - 5x - 3 \)[/tex].
- The coefficients are [tex]\( [3, -4, 3, -5, -3] \)[/tex].
2. Substitute [tex]\( x = -3 \)[/tex]:
- Begin with the first coefficient: [tex]\( 3 \)[/tex].
- Multiply by [tex]\( -3 \)[/tex], add the next coefficient:
- Step 1: [tex]\( 3 \)[/tex]
- Multiply: [tex]\( 3 \times (-3) = -9 \)[/tex]
- Add next coefficient: [tex]\( -9 + (-4) = -13 \)[/tex]
- Repeat the process:
- Step 2: [tex]\(-13 \times (-3) = 39\)[/tex], add 3: [tex]\( 39 + 3 = 42 \)[/tex]
- Step 3: [tex]\( 42 \times (-3) = -126\)[/tex], add (-5): [tex]\(-126 + (-5) = -131\)[/tex]
- Step 4: [tex]\(-131 \times (-3) = 393\)[/tex], add (-3): [tex]\(393 + (-3) = 390\)[/tex]
So, [tex]\( f(-3) = 390 \)[/tex].
### Synthetic Substitution for [tex]\( f(4) \)[/tex]:
1. Write down the coefficients again:
- Coefficients are [tex]\( [3, -4, 3, -5, -3] \)[/tex].
2. Substitute [tex]\( x = 4 \)[/tex]:
- Begin with the first coefficient: [tex]\( 3 \)[/tex].
- Multiply by [tex]\( 4 \)[/tex], add the next coefficient:
- Step 1: [tex]\( 3 \)[/tex]
- Multiply: [tex]\( 3 \times 4 = 12 \)[/tex]
- Add next coefficient: [tex]\( 12 + (-4) = 8 \)[/tex]
- Repeat the process:
- Step 2: [tex]\( 8 \times 4 = 32 \)[/tex], add 3: [tex]\( 32 + 3 = 35 \)[/tex]
- Step 3: [tex]\( 35 \times 4 = 140 \)[/tex], add (-5): [tex]\( 140 + (-5) = 135 \)[/tex]
- Step 4: [tex]\( 135 \times 4 = 540 \)[/tex], add (-3): [tex]\( 540 + (-3) = 537 \)[/tex]
So, [tex]\( f(4) = 537 \)[/tex].
Therefore, the values are [tex]\( f(-3) = 390 \)[/tex] and [tex]\( f(4) = 537 \)[/tex]. The correct answer is:
a. 390,537
### Synthetic Substitution for [tex]\( f(-3) \)[/tex]:
1. Write down the coefficients:
- The polynomial is [tex]\( 3x^4 - 4x^3 + 3x^2 - 5x - 3 \)[/tex].
- The coefficients are [tex]\( [3, -4, 3, -5, -3] \)[/tex].
2. Substitute [tex]\( x = -3 \)[/tex]:
- Begin with the first coefficient: [tex]\( 3 \)[/tex].
- Multiply by [tex]\( -3 \)[/tex], add the next coefficient:
- Step 1: [tex]\( 3 \)[/tex]
- Multiply: [tex]\( 3 \times (-3) = -9 \)[/tex]
- Add next coefficient: [tex]\( -9 + (-4) = -13 \)[/tex]
- Repeat the process:
- Step 2: [tex]\(-13 \times (-3) = 39\)[/tex], add 3: [tex]\( 39 + 3 = 42 \)[/tex]
- Step 3: [tex]\( 42 \times (-3) = -126\)[/tex], add (-5): [tex]\(-126 + (-5) = -131\)[/tex]
- Step 4: [tex]\(-131 \times (-3) = 393\)[/tex], add (-3): [tex]\(393 + (-3) = 390\)[/tex]
So, [tex]\( f(-3) = 390 \)[/tex].
### Synthetic Substitution for [tex]\( f(4) \)[/tex]:
1. Write down the coefficients again:
- Coefficients are [tex]\( [3, -4, 3, -5, -3] \)[/tex].
2. Substitute [tex]\( x = 4 \)[/tex]:
- Begin with the first coefficient: [tex]\( 3 \)[/tex].
- Multiply by [tex]\( 4 \)[/tex], add the next coefficient:
- Step 1: [tex]\( 3 \)[/tex]
- Multiply: [tex]\( 3 \times 4 = 12 \)[/tex]
- Add next coefficient: [tex]\( 12 + (-4) = 8 \)[/tex]
- Repeat the process:
- Step 2: [tex]\( 8 \times 4 = 32 \)[/tex], add 3: [tex]\( 32 + 3 = 35 \)[/tex]
- Step 3: [tex]\( 35 \times 4 = 140 \)[/tex], add (-5): [tex]\( 140 + (-5) = 135 \)[/tex]
- Step 4: [tex]\( 135 \times 4 = 540 \)[/tex], add (-3): [tex]\( 540 + (-3) = 537 \)[/tex]
So, [tex]\( f(4) = 537 \)[/tex].
Therefore, the values are [tex]\( f(-3) = 390 \)[/tex] and [tex]\( f(4) = 537 \)[/tex]. The correct answer is:
a. 390,537
