Simplifying Polynomial Expressions ES1 - Math Worksheets 4 Kids ... - Free Printable

To solve the given problems, we need to simplify each polynomial expression by combining like terms and performing the necessary operations (addition, subtraction, multiplication). Let's go through each problem step by step.

---

Problem 1:

\[
(2x^3 + 5x^2) - (12x + 2x^3)
\]

#### Solution:
1. Distribute the negative sign through the second parenthesis:
\[
2x^3 + 5x^2 - 12x - 2x^3
\]

2. Combine like terms:
- For \(x^3\) terms: \(2x^3 - 2x^3 = 0\)
- For \(x^2\) terms: \(5x^2\) (no other \(x^2\) terms)
- For \(x\) terms: \(-12x\) (no other \(x\) terms)

So, the simplified expression is:
\[
5x^2 - 12x
\]

#### Final Answer:
\[
\boxed{5x^2 - 12x}
\]

---

Problem 2:

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

#### Solution:
This is a multiplication of two polynomials. Use the distributive property (FOIL method for binomials):

1. Distribute each term in the first polynomial to each term in the second polynomial:
\[
(2x^2)(7x) + (2x^2)(6x^2) + (2x^4)(7x) + (2x^4)(6x^2)
\]

2. Perform the multiplications:
- \((2x^2)(7x) = 14x^3\)
- \((2x^2)(6x^2) = 12x^4\)
- \((2x^4)(7x) = 14x^5\)
- \((2x^4)(6x^2) = 12x^6\)

3. Combine all the terms:
\[
12x^6 + 14x^5 + 12x^4 + 14x^3
\]

#### Final Answer:
\[
\boxed{12x^6 + 14x^5 + 12x^4 + 14x^3}
\]

---

Problem 3:

\[
(12x^4 + x^2) - (2x^4 - 6x^2)
\]

#### Solution:
1. Distribute the negative sign through the second parenthesis:
\[
12x^4 + x^2 - 2x^4 + 6x^2
\]

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

So, the simplified expression is:
\[
10x^4 + 7x^2
\]

#### Final Answer:
\[
\boxed{10x^4 + 7x^2}
\]

---

Problem 4:

\[
14x + 3x^2 - 26x + 6x^2 + 7x
\]

#### Solution:
1. Combine like terms:
- For \(x^2\) terms: \(3x^2 + 6x^2 = 9x^2\)
- For \(x\) terms: \(14x - 26x + 7x = -5x\)

So, the simplified expression is:
\[
9x^2 - 5x
\]

#### Final Answer:
\[
\boxed{9x^2 - 5x}
\]

---

Problem 5:

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

#### Solution:
1. Distribute the 5 through the second parenthesis:
\[
5x^3 - 3 + 5x^2 - 15x
\]

2. Combine like terms:
- For \(x^3\) terms: \(5x^3\) (no other \(x^3\) terms)
- For \(x^2\) terms: \(5x^2\) (no other \(x^2\) terms)
- For \(x\) terms: \(-15x\) (no other \(x\) terms)
- For constant terms: \(-3\) (no other constants)

So, the simplified expression is:
\[
5x^3 + 5x^2 - 15x - 3
\]

#### Final Answer:
\[
\boxed{5x^3 + 5x^2 - 15x - 3}
\]

---

Problem 6:

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

#### Solution:
1. Distribute the \(-2\) through the second parenthesis:
\[
4x^3 - 2x - 2x^2 + 4x - 8
\]

2. Combine like terms:
- For \(x^3\) terms: \(4x^3\) (no other \(x^3\) terms)
- For \(x^2\) terms: \(-2x^2\) (no other \(x^2\) terms)
- For \(x\) terms: \(-2x + 4x = 2x\)
- For constant terms: \(-8\) (no other constants)

So, the simplified expression is:
\[
4x^3 - 2x^2 + 2x - 8
\]

#### Final Answer:
\[
\boxed{4x^3 - 2x^2 + 2x - 8}
\]

---

Problem 7:

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

#### Solution:
1. Distribute the constants through each parenthesis:
\[
2(4x^3) - 2(3x) - 3(2x^3) - 3(4x)
\]
\[
8x^3 - 6x - 6x^3 - 12x
\]

2. Combine like terms:
- For \(x^3\) terms: \(8x^3 - 6x^3 = 2x^3\)
- For \(x\) terms: \(-6x - 12x = -18x\)

So, the simplified expression is:
\[
2x^3 - 18x
\]

#### Final Answer:
\[
\boxed{2x^3 - 18x}
\]

---

Problem 8:

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

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

2. Combine like terms:
- For \(x^2\) terms: \(2x^2 - 2x^2 = 0\)
- For \(x\) terms: \(-2x - 5x = -7x\)

So, the simplified expression is:
\[
-7x
\]

#### Final Answer:
\[
\boxed{-7x}
\]

---

Problem 9:

\[
2x + (4x^2 + 2x) + x^2
\]

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

So, the simplified expression is:
\[
5x^2 + 4x
\]

#### Final Answer:
\[
\boxed{5x^2 + 4x}
\]

---

Problem 10:

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

#### Solution:
1. Distribute the \(-2\) through the parenthesis:
\[
x^2 - 2x^2 - 6x + 3x
\]

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

So, the simplified expression is:
\[
-x^2 - 3x
\]

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

---

Problem 11:

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

#### Solution:
1. Combine like terms:
- For \(x^3\) terms: \(-x^3\) (no other \(x^3\) terms)
- For \(x^2\) terms: \(2x^2 + 6x^2 = 8x^2\)
- For \(x\) terms: \(-x\) (no other \(x\) terms)

So, the simplified expression is:
\[
-x^3 + 8x^2 - x
\]

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

---

Problem 12:

\[
4x^2 - 3x^5 + 15x^3 - 12x
\]

#### Solution:
The terms are already in descending order of powers, and there are no like terms to combine. So, the expression is already simplified.

#### Final Answer:
\[
\boxed{-3x^5 + 4x^2 + 15x^3 - 12x}
\]

---

Problem 13:

\[
2x^2 + 5x^3 + 16x - 11x
\]

#### Solution:
1. Combine like terms:
- For \(x^3\) terms: \(5x^3\) (no other \(x^3\) terms)
- For \(x^2\) terms: \(2x^2\) (no other \(x^2\) terms)
- For \(x\) terms: \(16x - 11x = 5x\)

So, the simplified expression is:
\[
5x^3 + 2x^2 + 5x
\]

#### Final Answer:
\[
\boxed{5x^3 + 2x^2 + 5x}
\]

---

Problem 14:

\[
2x^2 + 5x^3 + 7x^2 + 12x
\]

#### Solution:
1. Combine like terms:
- For \(x^3\) terms: \(5x^3\) (no other \(x^3\) terms)
- For \(x^2\) terms: \(2x^2 + 7x^2 = 9x^2\)
- For \(x\) terms: \(12x\) (no other \(x\) terms)

So, the simplified expression is:
\[
5x^3 + 9x^2 + 12x
\]

#### Final Answer:
\[
\boxed{5x^3 + 9x^2 + 12x}
\]

---

Problem 15:

\[
2x^2 + 5x^3 + 8x
\]

#### Solution:
The terms are already in descending order of powers, and there are no like terms to combine. So, the expression is already simplified.

#### Final Answer:
\[
\boxed{5x^3 + 2x^2 + 8x}
\]

---

Problem 16:

\[
5x^2 + 15x - x^2 + 2x
\]

#### Solution:
1. Combine like terms:
- For \(x^2\) terms: \(5x^2 - x^2 = 4x^2\)
- For \(x\) terms: \(15x + 2x = 17x\)

So, the simplified expression is:
\[
4x^2 + 17x
\]

#### Final Answer:
\[
\boxed{4x^2 + 17x}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 5x^2 - 12x \\
2. & \ 12x^6 + 14x^5 + 12x^4 + 14x^3 \\
3. & \ 10x^4 + 7x^2 \\
4. & \ 9x^2 - 5x \\
5. & \ 5x^3 + 5x^2 - 15x - 3 \\
6. & \ 4x^3 - 2x^2 + 2x - 8 \\
7. & \ 2x^3 - 18x \\
8. & \ -7x \\
9. & \ 5x^2 + 4x \\
10. & \ -x^2 - 3x \\
11. & \ -x^3 + 8x^2 - x \\
12. & \ -3x^5 + 4x^2 + 15x^3 - 12x \\
13. & \ 5x^3 + 2x^2 + 5x \\
14. & \ 5x^3 + 9x^2 + 12x \\
15. & \ 5x^3 + 2x^2 + 8x \\
16. & \ 4x^2 + 17x \\
\end{aligned}
}
\]