Multiplying Polynomials worksheet with 15 practice problems.

To solve the problem of multiplying polynomials, we need to carefully apply the distributive property (also known as the FOIL method for binomials) and combine like terms. Let's go through each problem step by step.

---

Problem 1:

Multiply \( (3x^2)(-7x^4) \)

Solution:
- Multiply the coefficients: \( 3 \cdot (-7) = -21 \)
- Add the exponents of \( x \): \( x^2 \cdot x^4 = x^{2+4} = x^6 \)
- Result: \( -21x^6 \)

Answer: \( -21x^6 \)

---

Problem 2:

Multiply \( (4x^3)(8x^5) \)

Solution:
- Multiply the coefficients: \( 4 \cdot 8 = 32 \)
- Add the exponents of \( x \): \( x^3 \cdot x^5 = x^{3+5} = x^8 \)
- Result: \( 32x^8 \)

Answer: \( 32x^8 \)

---

Problem 3:

Multiply \( (3x^2)(-3x^3) \)

Solution:
- Multiply the coefficients: \( 3 \cdot (-3) = -9 \)
- Add the exponents of \( x \): \( x^2 \cdot x^3 = x^{2+3} = x^5 \)
- Result: \( -9x^5 \)

Answer: \( -9x^5 \)

---

Problem 4:

Multiply \( (3x + 5)(x - 3) \)

Solution:
- Use the distributive property (FOIL method):
\[
(3x + 5)(x - 3) = 3x \cdot x + 3x \cdot (-3) + 5 \cdot x + 5 \cdot (-3)
\]
\[
= 3x^2 - 9x + 5x - 15
\]
- Combine like terms: \( -9x + 5x = -4x \)
- Result: \( 3x^2 - 4x - 15 \)

Answer: \( 3x^2 - 4x - 15 \)

---

Problem 5:

Multiply \( (x + 6)(x + 8) \)

Solution:
- Use the distributive property (FOIL method):
\[
(x + 6)(x + 8) = x \cdot x + x \cdot 8 + 6 \cdot x + 6 \cdot 8
\]
\[
= x^2 + 8x + 6x + 48
\]
- Combine like terms: \( 8x + 6x = 14x \)
- Result: \( x^2 + 14x + 48 \)

Answer: \( x^2 + 14x + 48 \)

---

Problem 6:

Multiply \( (3x^2 - 2x + 5)(x + 3) \)

Solution:
- Distribute each term in the first polynomial to each term in the second polynomial:
\[
(3x^2 - 2x + 5)(x + 3) = 3x^2 \cdot x + 3x^2 \cdot 3 - 2x \cdot x - 2x \cdot 3 + 5 \cdot x + 5 \cdot 3
\]
\[
= 3x^3 + 9x^2 - 2x^2 - 6x + 5x + 15
\]
- Combine like terms: \( 9x^2 - 2x^2 = 7x^2 \) and \( -6x + 5x = -x \)
- Result: \( 3x^3 + 7x^2 - x + 15 \)

Answer: \( 3x^3 + 7x^2 - x + 15 \)

---

Problem 7:

Multiply \( (x + 8)(x - 2) \)

Solution:
- Use the distributive property (FOIL method):
\[
(x + 8)(x - 2) = x \cdot x + x \cdot (-2) + 8 \cdot x + 8 \cdot (-2)
\]
\[
= x^2 - 2x + 8x - 16
\]
- Combine like terms: \( -2x + 8x = 6x \)
- Result: \( x^2 + 6x - 16 \)

Answer: \( x^2 + 6x - 16 \)

---

Problem 8:

Multiply \( (x^2 - 6)(x - 2) \)

Solution:
- Distribute each term in the first polynomial to each term in the second polynomial:
\[
(x^2 - 6)(x - 2) = x^2 \cdot x + x^2 \cdot (-2) - 6 \cdot x - 6 \cdot (-2)
\]
\[
= x^3 - 2x^2 - 6x + 12
\]
- No like terms to combine.
- Result: \( x^3 - 2x^2 - 6x + 12 \)

Answer: \( x^3 - 2x^2 - 6x + 12 \)

---

Problem 9:

Multiply \( (x^2 - 3x + 5)(x^2 + 3x - 5) \)

Solution:
- Use the distributive property:
\[
(x^2 - 3x + 5)(x^2 + 3x - 5) = x^2(x^2 + 3x - 5) - 3x(x^2 + 3x - 5) + 5(x^2 + 3x - 5)
\]
\[
= x^4 + 3x^3 - 5x^2 - 3x^3 - 9x^2 + 15x + 5x^2 + 15x - 25
\]
- Combine like terms:
- \( x^4 \) (no other \( x^4 \) terms)
- \( 3x^3 - 3x^3 = 0 \)
- \( -5x^2 - 9x^2 + 5x^2 = -9x^2 \)
- \( 15x + 15x = 30x \)
- Constant term: \( -25 \)
- Result: \( x^4 - 9x^2 + 30x - 25 \)

Answer: \( x^4 - 9x^2 + 30x - 25 \)

---

Problem 10:

Multiply \( (x - 3)(x^2 + 3x - 5) \)

Solution:
- Distribute each term in the first polynomial to each term in the second polynomial:
\[
(x - 3)(x^2 + 3x - 5) = x(x^2 + 3x - 5) - 3(x^2 + 3x - 5)
\]
\[
= x^3 + 3x^2 - 5x - 3x^2 - 9x + 15
\]
- Combine like terms:
- \( x^3 \) (no other \( x^3 \) terms)
- \( 3x^2 - 3x^2 = 0 \)
- \( -5x - 9x = -14x \)
- Constant term: \( 15 \)
- Result: \( x^3 - 14x + 15 \)

Answer: \( x^3 - 14x + 15 \)

---

Problem 11:

Multiply \( (x + 4)(x^2 - 4x + 3) \)

Solution:
- Distribute each term in the first polynomial to each term in the second polynomial:
\[
(x + 4)(x^2 - 4x + 3) = x(x^2 - 4x + 3) + 4(x^2 - 4x + 3)
\]
\[
= x^3 - 4x^2 + 3x + 4x^2 - 16x + 12
\]
- Combine like terms:
- \( x^3 \) (no other \( x^3 \) terms)
- \( -4x^2 + 4x^2 = 0 \)
- \( 3x - 16x = -13x \)
- Constant term: \( 12 \)
- Result: \( x^3 - 13x + 12 \)

Answer: \( x^3 - 13x + 12 \)

---

Problem 12:

Multiply \( (x^2 + 6x)(x^2 - 6x) \)

Solution:
- Use the difference of squares formula: \( (a + b)(a - b) = a^2 - b^2 \)
\[
(x^2 + 6x)(x^2 - 6x) = (x^2)^2 - (6x)^2
\]
\[
= x^4 - 36x^2
\]
- Result: \( x^4 - 36x^2 \)

Answer: \( x^4 - 36x^2 \)

---

Problem 13:

Multiply \( (x^2 + 5x + 6)(x + 2) \)

Solution:
- Distribute each term in the first polynomial to each term in the second polynomial:
\[
(x^2 + 5x + 6)(x + 2) = x^2(x + 2) + 5x(x + 2) + 6(x + 2)
\]
\[
= x^3 + 2x^2 + 5x^2 + 10x + 6x + 12
\]
- Combine like terms:
- \( x^3 \) (no other \( x^3 \) terms)
- \( 2x^2 + 5x^2 = 7x^2 \)
- \( 10x + 6x = 16x \)
- Constant term: \( 12 \)
- Result: \( x^3 + 7x^2 + 16x + 12 \)

Answer: \( x^3 + 7x^2 + 16x + 12 \)

---

Problem 14:

Multiply \( (x^2 + 3x + 2)(x^2 - 3x + 2) \)

Solution:
- Use the distributive property:
\[
(x^2 + 3x + 2)(x^2 - 3x + 2) = x^2(x^2 - 3x + 2) + 3x(x^2 - 3x + 2) + 2(x^2 - 3x + 2)
\]
\[
= x^4 - 3x^3 + 2x^2 + 3x^3 - 9x^2 + 6x + 2x^2 - 6x + 4
\]
- Combine like terms:
- \( x^4 \) (no other \( x^4 \) terms)
- \( -3x^3 + 3x^3 = 0 \)
- \( 2x^2 - 9x^2 + 2x^2 = -5x^2 \)
- \( 6x - 6x = 0 \)
- Constant term: \( 4 \)
- Result: \( x^4 - 5x^2 + 4 \)

Answer: \( x^4 - 5x^2 + 4 \)

---

Problem 15:

Multiply \( (x + 5)(x^2 + 5x + 6) \)

Solution:
- Distribute each term in the first polynomial to each term in the second polynomial:
\[
(x + 5)(x^2 + 5x + 6) = x(x^2 + 5x + 6) + 5(x^2 + 5x + 6)
\]
\[
= x^3 + 5x^2 + 6x + 5x^2 + 25x + 30
\]
- Combine like terms:
- \( x^3 \) (no other \( x^3 \) terms)
- \( 5x^2 + 5x^2 = 10x^2 \)
- \( 6x + 25x = 31x \)
- Constant term: \( 30 \)
- Result: \( x^3 + 10x^2 + 31x + 30 \)

Answer: \( x^3 + 10x^2 + 31x + 30 \)

---

Final Answer:

\[
\boxed{
\begin{array}{ccc}
1. & -21x^6 & \\
2. & 32x^8 & \\
3. & -9x^5 & \\
4. & 3x^2 - 4x - 15 & \\
5. & x^2 + 14x + 48 & \\
6. & 3x^3 + 7x^2 - x + 15 & \\
7. & x^2 + 6x - 16 & \\
8. & x^3 - 2x^2 - 6x + 12 & \\
9. & x^4 - 9x^2 + 30x - 25 & \\
10. & x^3 - 14x + 15 & \\
11. & x^3 - 13x + 12 & \\
12. & x^4 - 36x^2 & \\
13. & x^3 + 7x^2 + 16x + 12 & \\
14. & x^4 - 5x^2 + 4 & \\
15. & x^3 + 10x^2 + 31x + 30 &
\end{array}
}
\]