Practice algebra skills by substituting values into polynomial equations with this comprehensive worksheet.

Problem: Evaluating Polynomial Functions

We will solve each part of the problem step by step.

---

#### Part A: Evaluate each function at the specified values.

1. Function: \( f(x) = 3x^6 + 2x^2 - 7x + 15 \); \( x = 1 \)

Substitute \( x = 1 \) into the function:
\[
f(1) = 3(1)^6 + 2(1)^2 - 7(1) + 15
\]
Simplify each term:
\[
f(1) = 3(1) + 2(1) - 7(1) + 15 = 3 + 2 - 7 + 15 = 13
\]

Answer: \( f(1) = 13 \)

2. Function: \( f(x) = x^5 \); \( x = -3 \)

Substitute \( x = -3 \) into the function:
\[
f(-3) = (-3)^5
\]
Calculate the power:
\[
(-3)^5 = -243
\]

Answer: \( f(-3) = -243 \)

---

#### Part B: Evaluate each function.

1. Function: \( f(x) = x^4 - 4x + 10 \); find \( f(4) \)

Substitute \( x = 4 \) into the function:
\[
f(4) = (4)^4 - 4(4) + 10
\]
Simplify each term:
\[
f(4) = 256 - 16 + 10 = 250
\]

Answer: \( f(4) = 250 \)

2. Function: \( f(x) = 5x^6 + 3x^3 - 2x^2 - 6 \); find \( f(-2) \)

Substitute \( x = -2 \) into the function:
\[
f(-2) = 5(-2)^6 + 3(-2)^3 - 2(-2)^2 - 6
\]
Calculate each term:
\[
(-2)^6 = 64, \quad (-2)^3 = -8, \quad (-2)^2 = 4
\]
Substitute these values:
\[
f(-2) = 5(64) + 3(-8) - 2(4) - 6
\]
Simplify:
\[
f(-2) = 320 - 24 - 8 - 6 = 282
\]

Answer: \( f(-2) = 282 \)

---

#### Part C: If \( f(x) = -x^3 - 8x^2 - 9x + 12 \); find the following.

1. Find \( f(3) \):
\[
f(3) = -(3)^3 - 8(3)^2 - 9(3) + 12
\]
Calculate each term:
\[
(3)^3 = 27, \quad (3)^2 = 9
\]
Substitute these values:
\[
f(3) = -27 - 8(9) - 9(3) + 12
\]
Simplify:
\[
f(3) = -27 - 72 - 27 + 12 = -114
\]

Answer: \( f(3) = -114 \)

2. Find \( f(-9) \):
\[
f(-9) = -(-9)^3 - 8(-9)^2 - 9(-9) + 12
\]
Calculate each term:
\[
(-9)^3 = -729, \quad (-9)^2 = 81
\]
Substitute these values:
\[
f(-9) = -(-729) - 8(81) - 9(-9) + 12
\]
Simplify:
\[
f(-9) = 729 - 648 + 81 + 12 = 174
\]

Answer: \( f(-9) = 174 \)

3. Find \( f(-7) \):
\[
f(-7) = -(-7)^3 - 8(-7)^2 - 9(-7) + 12
\]
Calculate each term:
\[
(-7)^3 = -343, \quad (-7)^2 = 49
\]
Substitute these values:
\[
f(-7) = -(-343) - 8(49) - 9(-7) + 12
\]
Simplify:
\[
f(-7) = 343 - 392 + 63 + 12 = 26
\]

Answer: \( f(-7) = 26 \)

4. Find \( f(0) \):
\[
f(0) = -(0)^3 - 8(0)^2 - 9(0) + 12
\]
Simplify:
\[
f(0) = 0 - 0 - 0 + 12 = 12
\]

Answer: \( f(0) = 12 \)

---

#### Part D: If \( f(x) = -x^4 + 5x^3 \); find the following.

1. Find \( f(7) + 3f(5) \):
- First, calculate \( f(7) \):
\[
f(7) = -(7)^4 + 5(7)^3
\]
Calculate each term:
\[
(7)^4 = 2401, \quad (7)^3 = 343
\]
Substitute these values:
\[
f(7) = -2401 + 5(343) = -2401 + 1715 = -686
\]
- Next, calculate \( f(5) \):
\[
f(5) = -(5)^4 + 5(5)^3
\]
Calculate each term:
\[
(5)^4 = 625, \quad (5)^3 = 125
\]
Substitute these values:
\[
f(5) = -625 + 5(125) = -625 + 625 = 0
\]
- Now, calculate \( f(7) + 3f(5) \):
\[
f(7) + 3f(5) = -686 + 3(0) = -686
\]

Answer: \( f(7) + 3f(5) = -686 \)

2. Find \( f(-3) - 7f(-2) \):
- First, calculate \( f(-3) \):
\[
f(-3) = -(-3)^4 + 5(-3)^3
\]
Calculate each term:
\[
(-3)^4 = 81, \quad (-3)^3 = -27
\]
Substitute these values:
\[
f(-3) = -81 + 5(-27) = -81 - 135 = -216
\]
- Next, calculate \( f(-2) \):
\[
f(-2) = -(-2)^4 + 5(-2)^3
\]
Calculate each term:
\[
(-2)^4 = 16, \quad (-2)^3 = -8
\]
Substitute these values:
\[
f(-2) = -16 + 5(-8) = -16 - 40 = -56
\]
- Now, calculate \( f(-3) - 7f(-2) \):
\[
f(-3) - 7f(-2) = -216 - 7(-56) = -216 + 392 = 176
\]

Answer: \( f(-3) - 7f(-2) = 176 \)

3. Find \( -f(1) \times f(4) \):
- First, calculate \( f(1) \):
\[
f(1) = -(1)^4 + 5(1)^3
\]
Calculate each term:
\[
(1)^4 = 1, \quad (1)^3 = 1
\]
Substitute these values:
\[
f(1) = -1 + 5(1) = -1 + 5 = 4
\]
- Next, calculate \( f(4) \):
\[
f(4) = -(4)^4 + 5(4)^3
\]
Calculate each term:
\[
(4)^4 = 256, \quad (4)^3 = 64
\]
Substitute these values:
\[
f(4) = -256 + 5(64) = -256 + 320 = 64
\]
- Now, calculate \( -f(1) \times f(4) \):
\[
-f(1) \times f(4) = -4 \times 64 = -256
\]

Answer: \( -f(1) \times f(4) = -256 \)

4. Find \( \frac{-f(-6)}{f(6)} \):
- First, calculate \( f(-6) \):
\[
f(-6) = -(-6)^4 + 5(-6)^3
\]
Calculate each term:
\[
(-6)^4 = 1296, \quad (-6)^3 = -216
\]
Substitute these values:
\[
f(-6) = -1296 + 5(-216) = -1296 - 1080 = -2376
\]
- Next, calculate \( f(6) \):
\[
f(6) = -(6)^4 + 5(6)^3
\]
Calculate each term:
\[
(6)^4 = 1296, \quad (6)^3 = 216
\]
Substitute these values:
\[
f(6) = -1296 + 5(216) = -1296 + 1080 = -216
\]
- Now, calculate \( \frac{-f(-6)}{f(6)} \):
\[
\frac{-f(-6)}{f(6)} = \frac{-(-2376)}{-216} = \frac{2376}{-216} = -11
\]

Answer: \( \frac{-f(-6)}{f(6)} = -11 \)

---

#### Part E: What is the value of \( f(3) \), if \( f(x) = x^6 - 14 \)?

Substitute \( x = 3 \) into the function:
\[
f(3) = (3)^6 - 14
\]
Calculate \( (3)^6 \):
\[
(3)^6 = 729
\]
Substitute this value:
\[
f(3) = 729 - 14 = 715
\]

Answer: \( f(3) = 715 \)

---

Final Answers:

\[
\boxed{
\begin{array}{l}
\text{A1: } 13 \\
\text{A2: } -243 \\
\text{B1: } 250 \\
\text{B2: } 282 \\
\text{C1: } -114 \\
\text{C2: } 174 \\
\text{C3: } 26 \\
\text{C4: } 12 \\
\text{D1: } -686 \\
\text{D2: } 176 \\
\text{D3: } -256 \\
\text{D4: } -11 \\
\text{E: } 715 \\
\end{array}
}
\]