Let's solve each of the four polynomial problems step by step, as shown in your image.
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(1) Add: $-2x^2 + 3xy + y^2$ and $-3x^2 - xy - 7y^2$
Step 1: Write both polynomials together:
$$
(-2x^2 + 3xy + y^2) + (-3x^2 - xy - 7y^2)
$$
Step 2: Combine like terms:
- $-2x^2 - 3x^2 = -5x^2$
- $3xy - xy = 2xy$
- $y^2 - 7y^2 = -6y^2$
Answer (1):
$$
\boxed{-5x^2 + 2xy - 6y^2}
$$
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(2) Subtract $7a - 3b - 5ab + c$ from $2a - 3b + c$
This means:
$$
(2a - 3b + c) - (7a - 3b - 5ab + c)
$$
Step 1: Distribute the negative sign:
$$
2a - 3b + c - 7a + 3b + 5ab - c
$$
Step 2: Combine like terms:
- $2a - 7a = -5a$
- $-3b + 3b = 0$
- $c - c = 0$
- $+5ab$
Answer (2):
$$
\boxed{-5a + 5ab}
$$
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(3) Multiply: $(2n + 3)$ and $(n - 2y)$
Use the distributive property (FOIL method):
$$
(2n + 3)(n - 2y) = 2n(n) + 2n(-2y) + 3(n) + 3(-2y)
$$
Now compute each term:
- $2n \cdot n = 2n^2$
- $2n \cdot (-2y) = -4ny$
- $3 \cdot n = 3n$
- $3 \cdot (-2y) = -6y$
Combine all:
$$
2n^2 - 4ny + 3n - 6y
$$
Answer (3):
$$
\boxed{2n^2 - 4ny + 3n - 6y}
$$
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(4) Divide: $2x^3 - 3x^2 + x - 2$ by $x$
We divide each term of the polynomial by $x$:
$$
\frac{2x^3}{x} - \frac{3x^2}{x} + \frac{x}{x} - \frac{2}{x}
$$
Simplify each term:
- $2x^3 / x = 2x^2$
- $-3x^2 / x = -3x$
- $x / x = 1$
- $-2 / x = -\frac{2}{x}$
So the result is:
$$
2x^2 - 3x + 1 - \frac{2}{x}
$$
Answer (4):
$$
\boxed{2x^2 - 3x + 1 - \frac{2}{x}}
$$
> Note: This is a rational expression because the last term has $x$ in the denominator. It’s not a polynomial anymore due to the $-\frac{2}{x}$ term.
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✔ Final Answers:
1. $-5x^2 + 2xy - 6y^2$
2. $-5a + 5ab$
3. $2n^2 - 4ny + 3n - 6y$
4. $2x^2 - 3x + 1 - \frac{2}{x}$
Let me know if you'd like these explained visually or with diagrams!
Parent Tip: Review the logic above to help your child master the concept of adding subtracting multiplying and dividing polynomials worksheet.
