Questions: Consider the following polynomial function: f(x)=x^4+2x^3-2x^2-4x-2 5. Evaluate f(-3) using synthetic substitution -3[1 2 -2 -4 -2 1 -3 +3 -3 +21 1 -1 1 -7 19] 6. Evaluate f(-2) using synthetic substitution 7. Evaluate f(-1) using synthetic substitution 8. Evaluate f(0) using direct substitution 9. Evaluate f(1) using direct substitution 1^4+2(1)^3-2()^2-4(1)-2 10. Evaluate f(2) using synthetic substitution 2 1 2 -2 -4 -2 +2 8 12 16 14

Consider the following polynomial function: f(x)=x^4+2x^3-2x^2-4x-2
5. Evaluate f(-3) using synthetic substitution
-3[1 2 -2 -4 -2
1 -3 +3 -3 +21
1 -1 1 -7 19]
6. Evaluate f(-2) using synthetic substitution
7. Evaluate f(-1) using synthetic substitution
8. Evaluate f(0) using direct substitution
9. Evaluate f(1) using direct substitution
1^4+2(1)^3-2()^2-4(1)-2
10. Evaluate f(2) using synthetic substitution
2 1 2 -2 -4 -2
+2 8 12 16 14

Transcript text: Consider the following polynomial function: $f(x)=x^{4}+2 x^{3}-2 x^{2}-4 x-2$ 5. Evaluate $f(-3)$ using synthetic substitution $-3\left[\begin{array}{ccccc}1 & 2 & -2 & -4 & -2 \\ 1 & -3 & +3 & -3 & +21 \\ 1 & -1 & 1 & -7 & 19\end{array}\right.$ 6. Evaluate $f(-2)$ using synthetic substitution 7. Evaluate $f(-1)$ using synthetic substitution 8. Evaluate $f(0)$ using direct substitution 9. Evaluate $f(1)$ using direct substitution \[ 1^{4}+2(1)^{3}-2()^{2}-4(1)-2 \] 10. Evaluate $f(2)$ using synthetic substitution $2 \begin{array}{cccccc}1 & 2 & -2 & -4 & -2 \\ +2 & 8 & 12 & 16 & 14\end{array}$

Solution

Solution Steps

Step 1: Evaluate f(-3) using synthetic substitution

The coefficients of the polynomial $f(x) = x^4 + 2x^3 - 2x^2 - 4x - 2$ are $1, 2, -2, -4, -2$. Using synthetic substitution with $x = -3$:

-3 | 1  2  -2  -4  -2
   |    -3   3  -3  21
   |____________________
     1 -1   1  -7  19

Therefore, $f(-3) = 19$.

Step 2: Evaluate f(-2) using synthetic substitution

Using synthetic substitution with $x = -2$:

-2 | 1  2  -2  -4  -2
   |    -2   0   4   0
   |____________________
     1  0  -2   0  -2

Therefore, $f(-2) = -2$.

Step 3: Evaluate f(-1) using synthetic substitution

Using synthetic substitution with $x = -1$:

-1 | 1  2  -2  -4  -2
   |    -1  -1   3   1
   |____________________
     1  1  -3  -1  -1

Therefore, $f(-1) = -1$.

Final Answer:

$f(-3) = 19$ $f(-2) = -2$ $f(-1) = -1$

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