Multiplying Polynomials worksheet with eight algebraic problems for practice.

Problem: Multiply the following polynomials.

We will solve each problem step by step using the distributive property, which states that for any expressions \(a\), \(b\), and \(c\):

\[
a(b + c) = ab + ac
\]

#### 1) \(3u^4(3u^2 + 5u - 7u^3 + 12)\)

Step 1: Distribute \(3u^4\) to each term inside the parentheses.

\[
3u^4 \cdot 3u^2 + 3u^4 \cdot 5u + 3u^4 \cdot (-7u^3) + 3u^4 \cdot 12
\]

Step 2: Multiply each term.

\[
(3 \cdot 3)u^{4+2} + (3 \cdot 5)u^{4+1} + (3 \cdot -7)u^{4+3} + (3 \cdot 12)u^4
\]

\[
9u^6 + 15u^5 - 21u^7 + 36u^4
\]

Step 3: Write the terms in descending order of the exponents.

\[
-21u^7 + 9u^6 + 15u^5 + 36u^4
\]

Final Answer:

\[
\boxed{-21u^7 + 9u^6 + 15u^5 + 36u^4}
\]

---

#### 2) \(-5x^2(-4x^8 + 8x^4 + 10)\)

Step 1: Distribute \(-5x^2\) to each term inside the parentheses.

\[
-5x^2 \cdot (-4x^8) + (-5x^2) \cdot 8x^4 + (-5x^2) \cdot 10
\]

Step 2: Multiply each term.

\[
(-5 \cdot -4)x^{2+8} + (-5 \cdot 8)x^{2+4} + (-5 \cdot 10)x^2
\]

\[
20x^{10} - 40x^6 - 50x^2
\]

Final Answer:

\[
\boxed{20x^{10} - 40x^6 - 50x^2}
\]

---

#### 3) \(7a^3(-4a - 6a^3)\)

Step 1: Distribute \(7a^3\) to each term inside the parentheses.

\[
7a^3 \cdot (-4a) + 7a^3 \cdot (-6a^3)
\]

Step 2: Multiply each term.

\[
(7 \cdot -4)a^{3+1} + (7 \cdot -6)a^{3+3}
\]

\[
-28a^4 - 42a^6
\]

Final Answer:

\[
\boxed{-42a^6 - 28a^4}
\]

---

#### 4) \(9v(-2v^3 + 6v^5 - 7v^2)\)

Step 1: Distribute \(9v\) to each term inside the parentheses.

\[
9v \cdot (-2v^3) + 9v \cdot 6v^5 + 9v \cdot (-7v^2)
\]

Step 2: Multiply each term.

\[
(9 \cdot -2)v^{1+3} + (9 \cdot 6)v^{1+5} + (9 \cdot -7)v^{1+2}
\]

\[
-18v^4 + 54v^6 - 63v^3
\]

Step 3: Write the terms in descending order of the exponents.

\[
54v^6 - 18v^4 - 63v^3
\]

Final Answer:

\[
\boxed{54v^6 - 18v^4 - 63v^3}
\]

---

#### 5) \(-s^5(1 + 17s^9)\)

Step 1: Distribute \(-s^5\) to each term inside the parentheses.

\[
-s^5 \cdot 1 + (-s^5) \cdot 17s^9
\]

Step 2: Multiply each term.

\[
-s^5 + (-1 \cdot 17)s^{5+9}
\]

\[
-s^5 - 17s^{14}
\]

Final Answer:

\[
\boxed{-17s^{14} - s^5}
\]

---

#### 6) \(8y^8(y^4 + 11)\)

Step 1: Distribute \(8y^8\) to each term inside the parentheses.

\[
8y^8 \cdot y^4 + 8y^8 \cdot 11
\]

Step 2: Multiply each term.

\[
(8 \cdot 1)y^{8+4} + (8 \cdot 11)y^8
\]

\[
8y^{12} + 88y^8
\]

Final Answer:

\[
\boxed{8y^{12} + 88y^8}
\]

---

#### 7) \(-b^2(4b^6 + 8b^4 + 2b + 9)\)

Step 1: Distribute \(-b^2\) to each term inside the parentheses.

\[
-b^2 \cdot 4b^6 + (-b^2) \cdot 8b^4 + (-b^2) \cdot 2b + (-b^2) \cdot 9
\]

Step 2: Multiply each term.

\[
(-1 \cdot 4)b^{2+6} + (-1 \cdot 8)b^{2+4} + (-1 \cdot 2)b^{2+1} + (-1 \cdot 9)b^2
\]

\[
-4b^8 - 8b^6 - 2b^3 - 9b^2
\]

Final Answer:

\[
\boxed{-4b^8 - 8b^6 - 2b^3 - 9b^2}
\]

---

#### 8) \(10(7m^5 - 3 - m^2)\)

Step 1: Distribute \(10\) to each term inside the parentheses.

\[
10 \cdot 7m^5 + 10 \cdot (-3) + 10 \cdot (-m^2)
\]

Step 2: Multiply each term.

\[
(10 \cdot 7)m^5 + (10 \cdot -3) + (10 \cdot -1)m^2
\]

\[
70m^5 - 30 - 10m^2
\]

Step 3: Write the terms in descending order of the exponents.

\[
70m^5 - 10m^2 - 30
\]

Final Answer:

\[
\boxed{70m^5 - 10m^2 - 30}
\]

---

Final Answers:

1. \(\boxed{-21u^7 + 9u^6 + 15u^5 + 36u^4}\)
2. \(\boxed{20x^{10} - 40x^6 - 50x^2}\)
3. \(\boxed{-42a^6 - 28a^4}\)
4. \(\boxed{54v^6 - 18v^4 - 63v^3}\)
5. \(\boxed{-17s^{14} - s^5}\)
6. \(\boxed{8y^{12} + 88y^8}\)
7. \(\boxed{-4b^8 - 8b^6 - 2b^3 - 9b^2}\)
8. \(\boxed{70m^5 - 10m^2 - 30}\)