To solve the problem of simplifying polynomial expressions, we need to combine like terms. Like terms are terms that have the same variable(s) raised to the same power(s). Let's go through each expression step by step.
---
Expression 1:
$$
-5x^2 - x^2 + 3x^2
$$
1. Identify the like terms: All terms involve $x^2$.
2. Combine the coefficients:
$$
-5x^2 - x^2 + 3x^2 = (-5 - 1 + 3)x^2 = -3x^2
$$
Simplified Expression:
$$
\boxed{-3x^2}
$$
---
Expression 2:
$$
-6x^2 + 2x^2 + x + 10x
$$
1. Group like terms:
- Terms involving $x^2$: $-6x^2 + 2x^2$
- Terms involving $x$: $x + 10x$
2. Combine the coefficients:
$$
-6x^2 + 2x^2 = (-6 + 2)x^2 = -4x^2
$$
$$
x + 10x = (1 + 10)x = 11x
$$
3. Write the simplified expression:
$$
-4x^2 + 11x
$$
Simplified Expression:
$$
\boxed{-4x^2 + 11x}
$$
---
Expression 3:
$$
10y^2 - 8y^2 + 4y^2 - 3y^2
$$
1. Identify the like terms: All terms involve $y^2$.
2. Combine the coefficients:
$$
10y^2 - 8y^2 + 4y^2 - 3y^2 = (10 - 8 + 4 - 3)y^2 = 3y^2
$$
Simplified Expression:
$$
\boxed{3y^2}
$$
---
Expression 4:
$$
-9x^2 + 7x^2 - 5x^2 + 3x^2
$$
1. Identify the like terms: All terms involve $x^2$.
2. Combine the coefficients:
$$
-9x^2 + 7x^2 - 5x^2 + 3x^2 = (-9 + 7 - 5 + 3)x^2 = -4x^2
$$
Simplified Expression:
$$
\boxed{-4x^2}
$$
---
Expression 5:
$$
-8x^2 - 5x^2 + 10x^2
$$
1. Identify the like terms: All terms involve $x^2$.
2. Combine the coefficients:
$$
-8x^2 - 5x^2 + 10x^2 = (-8 - 5 + 10)x^2 = -3x^2
$$
Simplified Expression:
$$
\boxed{-3x^2}
$$
---
Expression 6:
$$
m^2 - m^2 + 3m^2 - 2m^2
$$
1. Identify the like terms: All terms involve $m^2$.
2. Combine the coefficients:
$$
m^2 - m^2 + 3m^2 - 2m^2 = (1 - 1 + 3 - 2)m^2 = m^2
$$
Simplified Expression:
$$
\boxed{m^2}
$$
---
Expression 7:
$$
a^2d - a^2d - 3a^2d
$$
1. Identify the like terms: All terms involve $a^2d$.
2. Combine the coefficients:
$$
a^2d - a^2d - 3a^2d = (1 - 1 - 3)a^2d = -3a^2d
$$
Simplified Expression:
$$
\boxed{-3a^2d}
$$
---
Expression 8:
$$
10x^2 - 4x^2 + 6x^2 - 2x^2
$$
1. Identify the like terms: All terms involve $x^2$.
2. Combine the coefficients:
$$
10x^2 - 4x^2 + 6x^2 - 2x^2 = (10 - 4 + 6 - 2)x^2 = 10x^2
$$
Simplified Expression:
$$
\boxed{10x^2}
$$
---
Expression 9:
$$
10x^2 - 4x^2 + 6x^2 - 2x^2
$$
This is the same as Expression 8. The simplified expression is:
$$
\boxed{10x^2}
$$
---
Expression 10:
$$
10x^2 - 4x^2 + 6x^2 - 2x^2
$$
This is the same as Expression 8 and 9. The simplified expression is:
$$
\boxed{10x^2}
$$
---
Final Answer Summary:
$$
\boxed{
\begin{aligned}
1. & \quad -3x^2 \\
2. & \quad -4x^2 + 11x \\
3. & \quad 3y^2 \\
4. & \quad -4x^2 \\
5. & \quad -3x^2 \\
6. & \quad m^2 \\
7. & \quad -3a^2d \\
8. & \quad 10x^2 \\
9. & \quad 10x^2 \\
10. & \quad 10x^2 \\
\end{aligned}
}
$$
Parent Tip: Review the logic above to help your child master the concept of 7th grade algebraic expressions worksheet.
