Multiplying a Polynomial and a Monomial worksheet with ten practice problems for finding products.

To solve the problems of multiplying a polynomial by a monomial, we use the distributive property. The distributive property states that for any numbers \(a\), \(b\), and \(c\):

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

This means we multiply the monomial by each term in the polynomial separately and then sum the results. Let's solve each problem step by step.

---

Problem 1: \(3q(q^2 - 6q + 5)\)

Distribute \(3q\) to each term in the polynomial:

\[
3q \cdot q^2 + 3q \cdot (-6q) + 3q \cdot 5
\]

Simplify each term:

\[
3q^3 - 18q^2 + 15q
\]

So, the product is:

\[
\boxed{3q^3 - 18q^2 + 15q}
\]

---

Problem 2: \(9(x^2 + xy - 8y^2)\)

Distribute \(9\) to each term in the polynomial:

\[
9 \cdot x^2 + 9 \cdot xy + 9 \cdot (-8y^2)
\]

Simplify each term:

\[
9x^2 + 9xy - 72y^2
\]

So, the product is:

\[
\boxed{9x^2 + 9xy - 72y^2}
\]

---

Problem 3: \(7(6x^2 + 9xy + 10y^2)\)

Distribute \(7\) to each term in the polynomial:

\[
7 \cdot 6x^2 + 7 \cdot 9xy + 7 \cdot 10y^2
\]

Simplify each term:

\[
42x^2 + 63xy + 70y^2
\]

So, the product is:

\[
\boxed{42x^2 + 63xy + 70y^2}
\]

---

Problem 4: \(3x(2x^2 - 5x + 8)\)

Distribute \(3x\) to each term in the polynomial:

\[
3x \cdot 2x^2 + 3x \cdot (-5x) + 3x \cdot 8
\]

Simplify each term:

\[
6x^3 - 15x^2 + 24x
\]

So, the product is:

\[
\boxed{6x^3 - 15x^2 + 24x}
\]

---

Problem 5: \(x^2(-x^2 + 2x + 7)\)

Distribute \(x^2\) to each term in the polynomial:

\[
x^2 \cdot (-x^2) + x^2 \cdot 2x + x^2 \cdot 7
\]

Simplify each term:

\[
-x^4 + 2x^3 + 7x^2
\]

So, the product is:

\[
\boxed{-x^4 + 2x^3 + 7x^2}
\]

---

Problem 6: \(10(-x^2 + 10x - 6)\)

Distribute \(10\) to each term in the polynomial:

\[
10 \cdot (-x^2) + 10 \cdot 10x + 10 \cdot (-6)
\]

Simplify each term:

\[
-10x^2 + 100x - 60
\]

So, the product is:

\[
\boxed{-10x^2 + 100x - 60}
\]

---

Problem 7: \(2k^2(2k^3 + 6k - 4)\)

Distribute \(2k^2\) to each term in the polynomial:

\[
2k^2 \cdot 2k^3 + 2k^2 \cdot 6k + 2k^2 \cdot (-4)
\]

Simplify each term:

\[
4k^5 + 12k^3 - 8k^2
\]

So, the product is:

\[
\boxed{4k^5 + 12k^3 - 8k^2}
\]

---

Problem 8: \(6x^2(3x^2 - 5xy - 6)\)

Distribute \(6x^2\) to each term in the polynomial:

\[
6x^2 \cdot 3x^2 + 6x^2 \cdot (-5xy) + 6x^2 \cdot (-6)
\]

Simplify each term:

\[
18x^4 - 30x^3y - 36x^2
\]

So, the product is:

\[
\boxed{18x^4 - 30x^3y - 36x^2}
\]

---

Problem 9: \(-x(-x^2 - 4x + 4xy)\)

Distribute \(-x\) to each term in the polynomial:

\[
-x \cdot (-x^2) + (-x) \cdot (-4x) + (-x) \cdot 4xy
\]

Simplify each term:

\[
x^3 + 4x^2 - 4x^2y
\]

So, the product is:

\[
\boxed{x^3 + 4x^2 - 4x^2y}
\]

---

Problem 10: \(3r(2r^2 - 3ry + 2r)\)

Distribute \(3r\) to each term in the polynomial:

\[
3r \cdot 2r^2 + 3r \cdot (-3ry) + 3r \cdot 2r
\]

Simplify each term:

\[
6r^3 - 9r^2y + 6r^2
\]

So, the product is:

\[
\boxed{6r^3 - 9r^2y + 6r^2}
\]

---

Final Answers:

1. \(\boxed{3q^3 - 18q^2 + 15q}\)
2. \(\boxed{9x^2 + 9xy - 72y^2}\)
3. \(\boxed{42x^2 + 63xy + 70y^2}\)
4. \(\boxed{6x^3 - 15x^2 + 24x}\)
5. \(\boxed{-x^4 + 2x^3 + 7x^2}\)
6. \(\boxed{-10x^2 + 100x - 60}\)
7. \(\boxed{4k^5 + 12k^3 - 8k^2}\)
8. \(\boxed{18x^4 - 30x^3y - 36x^2}\)
9. \(\boxed{x^3 + 4x^2 - 4x^2y}\)
10. \(\boxed{6r^3 - 9r^2y + 6r^2}\)

Final Answer Boxed:

\[
\boxed{
\begin{aligned}
1. & \ 3q^3 - 18q^2 + 15q \\
2. & \ 9x^2 + 9xy - 72y^2 \\
3. & \ 42x^2 + 63xy + 70y^2 \\
4. & \ 6x^3 - 15x^2 + 24x \\
5. & \ -x^4 + 2x^3 + 7x^2 \\
6. & \ -10x^2 + 100x - 60 \\
7. & \ 4k^5 + 12k^3 - 8k^2 \\
8. & \ 18x^4 - 30x^3y - 36x^2 \\
9. & \ x^3 + 4x^2 - 4x^2y \\
10. & \ 6r^3 - 9r^2y + 6r^2
\end{aligned}
}
\]