Question

teacher :
adding and subtracting polynomials
simplify each expression.

1. $(6 + 9z^3) - (8z^3 - 5)$
2. $(7y^3 - 5y^2) + (8y^3 - 9y^2 + 2y^3)$
3. $(4s + 2 + 3s^3) - (6s^3 - 5 - 8s^3)$
4. $(3g - 2g^4) - (8g^4 + 9g)$
5. $(8n^4 - 4 - 7n^3) + (n + 3n^3 + 2)$
6. $(8p + 2q^2) - (7p + 4q^2 - 5)$
7. $(4m^2 - 8m) + (2m + 9m^2)$
8. $(4 - 6x^2) + (15x^2 - 7 + 3x^2)$
9. $(5 + 8a^3 - 8a^4) - (7a + 2a^2 - 4)$
10. $(2x + 5x^3) - (4x - 5x^3 + 3x^2)$

Explanation:

Step1: Simplify \( (6 + 9z^3) - (8z^2 - 5) \)

First, distribute the negative sign to the terms inside the second parentheses: \( 6 + 9z^3 - 8z^2 + 5 \).
Then, combine like terms (the constant terms): \( (6 + 5) + 9z^3 - 8z^2 = 11 + 9z^3 - 8z^2 \). Wait, the original handwritten answer is \( 11 + z^3 \)? Wait, maybe there was a typo in the problem. Wait, the problem is \( (6 + 9z^3) - (8z^2 - 5) \)? Wait, no, maybe the second term is \( (8z^3 - 5) \)? Let's check again. If the problem is \( (6 + 9z^3) - (8z^3 - 5) \), then:
Distribute the negative: \( 6 + 9z^3 - 8z^3 + 5 \).
Combine like terms: \( (9z^3 - 8z^3) + (6 + 5) = z^3 + 11 \), which matches the handwritten \( 11 + z^3 \). So probably a typo in the problem, the second term is \( 8z^3 - 5 \) instead of \( 8z^2 - 5 \). So proceeding with that correction.

Step2: Simplify \( (7y^3 - 5y^5) + (8y^3 - 9y^5 + 2y^4) \)

Combine like terms for \( y^5 \): \( -5y^5 - 9y^5 = -14y^5 \).
Combine like terms for \( y^3 \): \( 7y^3 + 8y^3 = 15y^3 \).
The \( y^4 \) term remains as \( 2y^4 \).
So the result is \( -14y^5 + 2y^4 + 15y^3 \).

Step3: Simplify \( (4s + 2 + 3s^3) - (6s^3 - 5 - 8s^2) \)

Distribute the negative sign: \( 4s + 2 + 3s^3 - 6s^3 + 5 + 8s^2 \).
Combine like terms for \( s^3 \): \( 3s^3 - 6s^3 = -3s^3 \).
Combine like terms for constants: \( 2 + 5 = 7 \).
So the result is \( -3s^3 + 8s^2 + 4s + 7 \).

Step4: Simplify \( (3g - 2g^4) - (8g^4 + 9g) \)

Distribute the negative sign: \( 3g - 2g^4 - 8g^4 - 9g \).
Combine like terms for \( g \): \( 3g - 9g = -6g \).
Combine like terms for \( g^4 \): \( -2g^4 - 8g^4 = -10g^4 \).
So the result is \( -10g^4 - 6g \).

Step5: Simplify \( (8n^4 - 4 - 7n^5) + (n + 3n^2 + 2) \)

Combine like terms for constants: \( -4 + 2 = -2 \).
The other terms: \( 8n^4 - 7n^5 + n + 3n^2 \).
Rearrange in descending order: \( -7n^5 + 8n^4 + 3n^2 + n - 2 \).

Step6: Simplify \( (8p + 2q^2) - (7p + 4q^2 - 5) \)

Distribute the negative sign: \( 8p + 2q^2 - 7p - 4q^2 + 5 \).
Combine like terms for \( p \): \( 8p - 7p = p \).
Combine like terms for \( q^2 \): \( 2q^2 - 4q^2 = -2q^2 \).
Combine constants: \( +5 \).
So the result is \( p - 2q^2 + 5 \).

Step7: Simplify \( (4m^2 - 8m) + (2m + 5m^2) \)

Combine like terms for \( m^2 \): \( 4m^2 + 5m^2 = 9m^2 \).
Combine like terms for \( m \): \( -8m + 2m = -6m \).
So the result is \( 9m^2 - 6m \).

Step8: Simplify \( (4 - 6x^2) + (15x^2 - 7 + 3x^5) \)

Combine like terms for \( x^2 \): \( -6x^2 + 15x^2 = 9x^2 \).
Combine constants: \( 4 - 7 = -3 \).
The \( x^5 \) term remains: \( 3x^5 \).
So the result is \( 3x^5 + 9x^2 - 3 \).

Step9: Simplify \( (5 + 8a^3 - 8a^4) - (7a + 2a^2 - 4) \)

Distribute the negative sign: \( 5 + 8a^3 - 8a^4 - 7a - 2a^2 + 4 \).
Combine constants: \( 5 + 4 = 9 \).
The other terms: \( -8a^4 + 8a^3 - 2a^2 - 7a + 9 \).

Step10: Simplify \( (2x + 5x^3) - (4x - 5x^3 + 3x^2) \)

Distribute the negative sign: \( 2x + 5x^3 - 4x + 5x^3 - 3x^2 \).
Combine like terms for \( x \): \( 2x - 4x = -2x \).
Combine like terms for \( x^3 \): \( 5x^3 + 5x^3 = 10x^3 \).
The \( x^2 \) term: \( -3x^2 \).
So the result is \( 10x^3 - 3x^2 - 2x \).

Answer:

s (for each problem):

1. \( 11 + z^3 \) (assuming the problem had a typo, \( 8z^3 \) instead of \( 8z^2 \))
2. \( -14y^5 + 2y^4 + 15y^3 \)
3. \( -3s^3 + 8s^2 + 4s + 7 \)
4. \( -10g^4 - 6g \)
5. \( -7n^5 + 8n^4 + 3n^2 + n - 2 \)
6. \( p - 2q^2 + 5 \)
7. \( 9m^2 - 6m \)
8. \( 3x^5 + 9x^2 - 3 \)
9. \( -8a^4 + 8a^3 - 2a^2 - 7a + 9 \)
10. \( 10x^3 - 3x^2 - 2x \)