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Show Answer Key & Explanations Step-by-step solution for: SOLUTION: Adding and subtracting polynomials - Studypool

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The task involves simplifying algebraic expressions. Let's go through each problem step by step.

---

Problem 1:

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

#### Solution:
1. Distribute the negative sign through the second parenthesis:
\[
2x^2 - 3x - 5x^2 - x
\]

2. Combine like terms:
- For \(x^2\) terms: \(2x^2 - 5x^2 = -3x^2\)
- For \(x\) terms: \(-3x - x = -4x\)

3. Write the simplified expression:
\[
-3x^2 - 4x
\]

Answer:
\[
\boxed{-3x^2 - 4x}
\]

---

Problem 2:

\[
(3x^2 - 4x^3 - 6x) + (7x^2 - x^3 - 8x)
\]

#### Solution:
1. Combine like terms:
- For \(x^3\) terms: \(-4x^3 - x^3 = -5x^3\)
- For \(x^2\) terms: \(3x^2 + 7x^2 = 10x^2\)
- For \(x\) terms: \(-6x - 8x = -14x\)

2. Write the simplified expression:
\[
-5x^3 + 10x^2 - 14x
\]

Answer:
\[
\boxed{-5x^3 + 10x^2 - 14x}
\]

---

Problem 3:

\[
(4x^2 - 7) - (3x^2 - 4x^3 - 6x)
\]

#### Solution:
1. Distribute the negative sign through the second parenthesis:
\[
4x^2 - 7 - 3x^2 + 4x^3 + 6x
\]

2. Combine like terms:
- For \(x^3\) terms: \(4x^3\) (only one term)
- For \(x^2\) terms: \(4x^2 - 3x^2 = x^2\)
- For \(x\) terms: \(6x\) (only one term)
- For constant terms: \(-7\) (only one term)

3. Write the simplified expression:
\[
4x^3 + x^2 + 6x - 7
\]

Answer:
\[
\boxed{4x^3 + x^2 + 6x - 7}
\]

---

Problem 4:

\[
(5x^2 - 3x - 4) - (2x^2 + 3x - 7)
\]

#### Solution:
1. Distribute the negative sign through the second parenthesis:
\[
5x^2 - 3x - 4 - 2x^2 - 3x + 7
\]

2. Combine like terms:
- For \(x^2\) terms: \(5x^2 - 2x^2 = 3x^2\)
- For \(x\) terms: \(-3x - 3x = -6x\)
- For constant terms: \(-4 + 7 = 3\)

3. Write the simplified expression:
\[
3x^2 - 6x + 3
\]

Answer:
\[
\boxed{3x^2 - 6x + 3}
\]

---

Problem 5:

\[
(6x - 7) - (4x - 9)
\]

#### Solution:
1. Distribute the negative sign through the second parenthesis:
\[
6x - 7 - 4x + 9
\]

2. Combine like terms:
- For \(x\) terms: \(6x - 4x = 2x\)
- For constant terms: \(-7 + 9 = 2\)

3. Write the simplified expression:
\[
2x + 2
\]

Answer:
\[
\boxed{2x + 2}
\]

---

Problem 6:

\[
(7x^2 - 3x^3 - 5x) + (2x^2 - 4x^3 - 6x)
\]

#### Solution:
1. Combine like terms:
- For \(x^3\) terms: \(-3x^3 - 4x^3 = -7x^3\)
- For \(x^2\) terms: \(7x^2 + 2x^2 = 9x^2\)
- For \(x\) terms: \(-5x - 6x = -11x\)

2. Write the simplified expression:
\[
-7x^3 + 9x^2 - 11x
\]

Answer:
\[
\boxed{-7x^3 + 9x^2 - 11x}
\]

---

Problem 7:

\[
(8x^2 - 3x - 4) + (2x^2 + 3x - 7)
\]

#### Solution:
1. Combine like terms:
- For \(x^2\) terms: \(8x^2 + 2x^2 = 10x^2\)
- For \(x\) terms: \(-3x + 3x = 0\)
- For constant terms: \(-4 - 7 = -11\)

2. Write the simplified expression:
\[
10x^2 - 11
\]

Answer:
\[
\boxed{10x^2 - 11}
\]

---

Problem 8:

\[
(9x^2 - 3x^3 - 5x) + (2x^2 - 4x^3 - 6x)
\]

#### Solution:
1. Combine like terms:
- For \(x^3\) terms: \(-3x^3 - 4x^3 = -7x^3\)
- For \(x^2\) terms: \(9x^2 + 2x^2 = 11x^2\)
- For \(x\) terms: \(-5x - 6x = -11x\)

2. Write the simplified expression:
\[
-7x^3 + 11x^2 - 11x
\]

Answer:
\[
\boxed{-7x^3 + 11x^2 - 11x}
\]

---

Problem 9:

\[
(10x^2 - 3x - 4) - (2x^2 + 3x - 7)
\]

#### Solution:
1. Distribute the negative sign through the second parenthesis:
\[
10x^2 - 3x - 4 - 2x^2 - 3x + 7
\]

2. Combine like terms:
- For \(x^2\) terms: \(10x^2 - 2x^2 = 8x^2\)
- For \(x\) terms: \(-3x - 3x = -6x\)
- For constant terms: \(-4 + 7 = 3\)

3. Write the simplified expression:
\[
8x^2 - 6x + 3
\]

Answer:
\[
\boxed{8x^2 - 6x + 3}
\]

---

Problem 10:

\[
(11x^2 - 3x^3 - 5x) + (2x^2 - 4x^3 - 6x)
\]

#### Solution:
1. Combine like terms:
- For \(x^3\) terms: \(-3x^3 - 4x^3 = -7x^3\)
- For \(x^2\) terms: \(11x^2 + 2x^2 = 13x^2\)
- For \(x\) terms: \(-5x - 6x = -11x\)

2. Write the simplified expression:
\[
-7x^3 + 13x^2 - 11x
\]

Answer:
\[
\boxed{-7x^3 + 13x^2 - 11x}
\]

---

Problem 11:

\[
(12x^2 - 3x - 4) - (2x^2 + 3x - 7)
\]

#### Solution:
1. Distribute the negative sign through the second parenthesis:
\[
12x^2 - 3x - 4 - 2x^2 - 3x + 7
\]

2. Combine like terms:
- For \(x^2\) terms: \(12x^2 - 2x^2 = 10x^2\)
- For \(x\) terms: \(-3x - 3x = -6x\)
- For constant terms: \(-4 + 7 = 3\)

3. Write the simplified expression:
\[
10x^2 - 6x + 3
\]

Answer:
\[
\boxed{10x^2 - 6x + 3}
\]

---

Problem 12:

\[
(13x^2 - 3x^3 - 5x) + (2x^2 - 4x^3 - 6x)
\]

#### Solution:
1. Combine like terms:
- For \(x^3\) terms: \(-3x^3 - 4x^3 = -7x^3\)
- For \(x^2\) terms: \(13x^2 + 2x^2 = 15x^2\)
- For \(x\) terms: \(-5x - 6x = -11x\)

2. Write the simplified expression:
\[
-7x^3 + 15x^2 - 11x
\]

Answer:
\[
\boxed{-7x^3 + 15x^2 - 11x}
\]

---

Final Answer:

\[
\boxed{
\begin{aligned}
&1. -3x^2 - 4x \\
&2. -5x^3 + 10x^2 - 14x \\
&3. 4x^3 + x^2 + 6x - 7 \\
&4. 3x^2 - 6x + 3 \\
&5. 2x + 2 \\
&6. -7x^3 + 9x^2 - 11x \\
&7. 10x^2 - 11 \\
&8. -7x^3 + 11x^2 - 11x \\
&9. 8x^2 - 6x + 3 \\
&10. -7x^3 + 13x^2 - 11x \\
&11. 10x^2 - 6x + 3 \\
&12. -7x^3 + 15x^2 - 11x
\end{aligned}
}
\]

Parent Tip: Review the logic above to help your child master the concept of algebra 1 adding and subtracting polynomials worksheet.

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