Problem: Add the Polynomials
The task involves adding pairs of polynomials. Let's solve each problem step by step.
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####
Problem 1:
Add:
$$
(2x + 6) + (2x)
$$
Step 1: Write down the polynomials.
$$
2x + 6 \quad \text{and} \quad 2x
$$
Step 2: Combine like terms.
- The \(x\)-terms are \(2x\) and \(2x\).
- The constant term is \(6\).
$$
(2x + 2x) + 6 = 4x + 6
$$
Final Answer for Problem 1:
$$
\boxed{4x + 6}
$$
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####
Problem 2:
Add:
$$
(x^2 + 5x - 4) + (3x^2 - 3x + 4)
$$
Step 1: Write down the polynomials.
$$
x^2 + 5x - 4 \quad \text{and} \quad 3x^2 - 3x + 4
$$
Step 2: Combine like terms.
- The \(x^2\)-terms are \(x^2\) and \(3x^2\).
- The \(x\)-terms are \(5x\) and \(-3x\).
- The constant terms are \(-4\) and \(4\).
$$
(x^2 + 3x^2) + (5x - 3x) + (-4 + 4) = 4x^2 + 2x + 0
$$
Final Answer for Problem 2:
$$
\boxed{4x^2 + 2x}
$$
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####
Problem 3:
Add:
$$
(5x^2 + 4xy + 7y^2) + (3x^2 - 7xy - 5y^2)
$$
Step 1: Write down the polynomials.
$$
5x^2 + 4xy + 7y^2 \quad \text{and} \quad 3x^2 - 7xy - 5y^2
$$
Step 2: Combine like terms.
- The \(x^2\)-terms are \(5x^2\) and \(3x^2\).
- The \(xy\)-terms are \(4xy\) and \(-7xy\).
- The \(y^2\)-terms are \(7y^2\) and \(-5y^2\).
$$
(5x^2 + 3x^2) + (4xy - 7xy) + (7y^2 - 5y^2) = 8x^2 - 3xy + 2y^2
$$
Final Answer for Problem 3:
$$
\boxed{8x^2 - 3xy + 2y^2}
$$
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####
Problem 4:
Add:
$$
(4k^4 - 8k^3 + 6k^2) + (2k^4 - 9k^3 - 5k^2)
$$
Step 1: Write down the polynomials.
$$
4k^4 - 8k^3 + 6k^2 \quad \text{and} \quad 2k^4 - 9k^3 - 5k^2
$$
Step 2: Combine like terms.
- The \(k^4\)-terms are \(4k^4\) and \(2k^4\).
- The \(k^3\)-terms are \(-8k^3\) and \(-9k^3\).
- The \(k^2\)-terms are \(6k^2\) and \(-5k^2\).
$$
(4k^4 + 2k^4) + (-8k^3 - 9k^3) + (6k^2 - 5k^2) = 6k^4 - 17k^3 + k^2
$$
Final Answer for Problem 4:
$$
\boxed{6k^4 - 17k^3 + k^2}
$$
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####
Problem 5:
Add:
$$
(6k - k^2 + c) + (8k^2 + 2k + 7c)
$$
Step 1: Write down the polynomials.
$$
6k - k^2 + c \quad \text{and} \quad 8k^2 + 2k + 7c
$$
Step 2: Combine like terms.
- The \(k^2\)-terms are \(-k^2\) and \(8k^2\).
- The \(k\)-terms are \(6k\) and \(2k\).
- The \(c\)-terms are \(c\) and \(7c\).
$$
(-k^2 + 8k^2) + (6k + 2k) + (c + 7c) = 7k^2 + 8k + 8c
$$
Final Answer for Problem 5:
$$
\boxed{7k^2 + 8k + 8c}
$$
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####
Problem 6:
Add:
$$
(8a^2 + 6a - 8) + (4a^2 + 6a - 8)
$$
Step 1: Write down the polynomials.
$$
8a^2 + 6a - 8 \quad \text{and} \quad 4a^2 + 6a - 8
$$
Step 2: Combine like terms.
- The \(a^2\)-terms are \(8a^2\) and \(4a^2\).
- The \(a\)-terms are \(6a\) and \(6a\).
- The constant terms are \(-8\) and \(-8\).
$$
(8a^2 + 4a^2) + (6a + 6a) + (-8 - 8) = 12a^2 + 12a - 16
$$
Final Answer for Problem 6:
$$
\boxed{12a^2 + 12a - 16}
$$
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Final Answers:
1. \(\boxed{4x + 6}\)
2. \(\boxed{4x^2 + 2x}\)
3. \(\boxed{8x^2 - 3xy + 2y^2}\)
4. \(\boxed{6k^4 - 17k^3 + k^2}\)
5. \(\boxed{7k^2 + 8k + 8c}\)
6. \(\boxed{12a^2 + 12a - 16}\)
Parent Tip: Review the logic above to help your child master the concept of adding polynomials worksheet answers.
