Multiply And Divide Functions - Worksheet - Free Printable

Problem: Multiply and Divide Functions

The task involves finding the product of two functions, \( (f \cdot g)(x) \), for various pairs of functions. Let's solve each problem step by step.

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#### Problem 1:
Find \( (f \cdot g)(m) \)
\[
f(m) = 4m^2 - m + 6
\]
\[
g(m) = -17m^5 - 7m^3
\]

Solution:
To find \( (f \cdot g)(m) \), we multiply \( f(m) \) and \( g(m) \):
\[
(f \cdot g)(m) = f(m) \cdot g(m) = (4m^2 - m + 6)(-17m^5 - 7m^3)
\]

We use the distributive property to expand this product:
\[
(f \cdot g)(m) = (4m^2)(-17m^5) + (4m^2)(-7m^3) + (-m)(-17m^5) + (-m)(-7m^3) + (6)(-17m^5) + (6)(-7m^3)
\]

Now, compute each term:
1. \( (4m^2)(-17m^5) = -68m^7 \)
2. \( (4m^2)(-7m^3) = -28m^5 \)
3. \( (-m)(-17m^5) = 17m^6 \)
4. \( (-m)(-7m^3) = 7m^4 \)
5. \( (6)(-17m^5) = -102m^5 \)
6. \( (6)(-7m^3) = -42m^3 \)

Combine all the terms:
\[
(f \cdot g)(m) = -68m^7 + 17m^6 - 28m^5 - 102m^5 + 7m^4 - 42m^3
\]

Simplify by combining like terms:
\[
(f \cdot g)(m) = -68m^7 + 17m^6 - 130m^5 + 7m^4 - 42m^3
\]

Final Answer:
\[
\boxed{-68m^7 + 17m^6 - 130m^5 + 7m^4 - 42m^3}
\]

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#### Problem 2:
Find \( (f \cdot g)(n) \)
\[
f(n) = n^4 - 2n + 4
\]
\[
g(n) = -10n^4 - 16n^3
\]

Solution:
To find \( (f \cdot g)(n) \), we multiply \( f(n) \) and \( g(n) \):
\[
(f \cdot g)(n) = f(n) \cdot g(n) = (n^4 - 2n + 4)(-10n^4 - 16n^3)
\]

We use the distributive property to expand this product:
\[
(f \cdot g)(n) = (n^4)(-10n^4) + (n^4)(-16n^3) + (-2n)(-10n^4) + (-2n)(-16n^3) + (4)(-10n^4) + (4)(-16n^3)
\]

Now, compute each term:
1. \( (n^4)(-10n^4) = -10n^8 \)
2. \( (n^4)(-16n^3) = -16n^7 \)
3. \( (-2n)(-10n^4) = 20n^5 \)
4. \( (-2n)(-16n^3) = 32n^4 \)
5. \( (4)(-10n^4) = -40n^4 \)
6. \( (4)(-16n^3) = -64n^3 \)

Combine all the terms:
\[
(f \cdot g)(n) = -10n^8 - 16n^7 + 20n^5 + 32n^4 - 40n^4 - 64n^3
\]

Simplify by combining like terms:
\[
(f \cdot g)(n) = -10n^8 - 16n^7 + 20n^5 - 8n^4 - 64n^3
\]

Final Answer:
\[
\boxed{-10n^8 - 16n^7 + 20n^5 - 8n^4 - 64n^3}
\]

---

#### Problem 3:
Find \( (f \cdot g)(z) \)
\[
f(z) = -3z^5 + 2z - 1
\]
\[
g(z) = -5z^2 - 14
\]

Solution:
To find \( (f \cdot g)(z) \), we multiply \( f(z) \) and \( g(z) \):
\[
(f \cdot g)(z) = f(z) \cdot g(z) = (-3z^5 + 2z - 1)(-5z^2 - 14)
\]

We use the distributive property to expand this product:
\[
(f \cdot g)(z) = (-3z^5)(-5z^2) + (-3z^5)(-14) + (2z)(-5z^2) + (2z)(-14) + (-1)(-5z^2) + (-1)(-14)
\]

Now, compute each term:
1. \( (-3z^5)(-5z^2) = 15z^7 \)
2. \( (-3z^5)(-14) = 42z^5 \)
3. \( (2z)(-5z^2) = -10z^3 \)
4. \( (2z)(-14) = -28z \)
5. \( (-1)(-5z^2) = 5z^2 \)
6. \( (-1)(-14) = 14 \)

Combine all the terms:
\[
(f \cdot g)(z) = 15z^7 + 42z^5 - 10z^3 + 5z^2 - 28z + 14
\]

Final Answer:
\[
\boxed{15z^7 + 42z^5 - 10z^3 + 5z^2 - 28z + 14}
\]

---

#### Problem 4:
Find \( (f \cdot g)(2) \)
\[
f(z) = -8z^2 - 16z - 2
\]
\[
g(z) = -z^3 + 3z^2 + 4
\]

Solution:
First, find \( f(2) \) and \( g(2) \):
\[
f(2) = -8(2)^2 - 16(2) - 2 = -8(4) - 32 - 2 = -32 - 32 - 2 = -66
\]
\[
g(2) = -(2)^3 + 3(2)^2 + 4 = -8 + 3(4) + 4 = -8 + 12 + 4 = 8
\]

Now, compute \( (f \cdot g)(2) \):
\[
(f \cdot g)(2) = f(2) \cdot g(2) = (-66)(8) = -528
\]

Final Answer:
\[
\boxed{-528}
\]

---

#### Problem 5:
Find \( (f \cdot g)(2) \)
\[
f(b) = -3b^3 - 3b - 18
\]
\[
g(b) = 4b^4 + 3b^3 + 5
\]

Solution:
First, find \( f(2) \) and \( g(2) \):
\[
f(2) = -3(2)^3 - 3(2) - 18 = -3(8) - 6 - 18 = -24 - 6 - 18 = -48
\]
\[
g(2) = 4(2)^4 + 3(2)^3 + 5 = 4(16) + 3(8) + 5 = 64 + 24 + 5 = 93
\]

Now, compute \( (f \cdot g)(2) \):
\[
(f \cdot g)(2) = f(2) \cdot g(2) = (-48)(93) = -4464
\]

Final Answer:
\[
\boxed{-4464}
\]

---

#### Problem 6:
Find \( (f \cdot g)(3) \)
\[
f(x) = -18x^4 - 19x^2 - 17
\]
\[
g(x) = 4x^4 + 6x^3 + 2x^2
\]

Solution:
First, find \( f(3) \) and \( g(3) \):
\[
f(3) = -18(3)^4 - 19(3)^2 - 17 = -18(81) - 19(9) - 17 = -1458 - 171 - 17 = -1646
\]
\[
g(3) = 4(3)^4 + 6(3)^3 + 2(3)^2 = 4(81) + 6(27) + 2(9) = 324 + 162 + 18 = 504
\]

Now, compute \( (f \cdot g)(3) \):
\[
(f \cdot g)(3) = f(3) \cdot g(3) = (-1646)(504) = -829,704
\]

Final Answer:
\[
\boxed{-829704}
\]

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Final Answers:

1. \(\boxed{-68m^7 + 17m^6 - 130m^5 + 7m^4 - 42m^3}\)
2. \(\boxed{-10n^8 - 16n^7 + 20n^5 - 8n^4 - 64n^3}\)
3. \(\boxed{15z^7 + 42z^5 - 10z^3 + 5z^2 - 28z + 14}\)
4. \(\boxed{-528}\)
5. \(\boxed{-4464}\)
6. \(\boxed{-829704}\)