Algebra II class notes covering polynomial operations, including addition, subtraction, and multiplication, with handwritten examples and solutions.

Let's go through the Algebra II - Class Notes on Adding, Subtracting, and Multiplying Polynomials, step by step. I'll solve each example and explain the process clearly.

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Example 1 – Add Horizontally

$$
(4x^3 + 2x^2 - 5x - 9) + (x^3 - 4x^2 - 8)
$$

Step-by-step:

We combine like terms:
- $4x^3 + x^3 = 5x^3$
- $2x^2 - 4x^2 = -2x^2$
- $-5x$ (no other $x$ term)
- $-9 - 8 = -17$

✔ Final Answer:
$$
\boxed{5x^3 - 2x^2 - 5x - 17}
$$

> ✔️ Your handwritten answer is correct!

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Example 2 – Add Vertically

$$
(8x^3 + 7x + 2) + (5x^2 - 3x + 1)
$$

Step-by-step:

Align like terms vertically:

```
8x^3 + 0x^2 + 7x + 2
+ 0x^3 + 5x^2 - 3x + 1
------------------------
8x^3 + 5x^2 + 4x + 3
```

- $8x^3 + 0x^3 = 8x^3$
- $0x^2 + 5x^2 = 5x^2$
- $7x - 3x = 4x$
- $2 + 1 = 3$

✔ Final Answer:
$$
\boxed{8x^3 + 5x^2 + 4x + 3}
$$

> ✔️ Your work is correct!

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Example 3 – Subtract Horizontally

$$
(2x^4 - 3x^3 - 7x^2 + 2x + 1) - (6x^4 + 2x^3 + 3x - 5)
$$

Step-by-step:

Distribute the negative sign to the second polynomial:

$$
= 2x^4 - 3x^3 - 7x^2 + 2x + 1 - 6x^4 - 2x^3 - 3x + 5
$$

Now combine like terms:
- $2x^4 - 6x^4 = -4x^4$
- $-3x^3 - 2x^3 = -5x^3$
- $-7x^2$ (no other $x^2$ term)
- $2x - 3x = -x$
- $1 + 5 = 6$

✔ Final Answer:
$$
\boxed{-4x^4 - 5x^3 - 7x^2 - x + 6}
$$

> ✔️ Your answer is correct!

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Example 4 – Subtract Vertically

$$
(5x^4 - 2x^3 + 6x + 4) - (7x^4 + 3x^2 + 8)
$$

Write it vertically with subtraction:

```
5x^4 - 2x^3 + 0x^2 + 6x + 4
- 7x^4 + 0x^3 + 3x^2 + 0x + 8
-------------------------------
-2x^4 - 2x^3 - 3x^2 + 6x - 4
```

Wait! Let's double-check this:

- $5x^4 - 7x^4 = -2x^4$
- $-2x^3 - 0x^3 = -2x^3$
- $0x^2 - 3x^2 = -3x^2$
- $6x - 0x = 6x$
- $4 - 8 = -4$

✔ Final Answer:
$$
\boxed{-2x^4 - 2x^3 - 3x^2 + 6x - 4}
$$

> ✘ You wrote $-2x^4 - 2x^3 - 3x^2 + 6x + 14$, but the constant should be $-4$, not $+14$.
> ✔ Corrected: $\boxed{-2x^4 - 2x^3 - 3x^2 + 6x - 4}$

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Example 5 – Multiply

#### a. $4x^2(3x^3 + 3x^2 - 2x - 4)$

Distribute $4x^2$ to each term:

- $4x^2 \cdot 3x^3 = 12x^5$
- $4x^2 \cdot 3x^2 = 12x^4$
- $4x^2 \cdot (-2x) = -8x^3$
- $4x^2 \cdot (-4) = -16x^2$

✔ Final Answer:
$$
\boxed{12x^5 + 12x^4 - 8x^3 - 16x^2}
$$

> ✔️ Your answer is correct!

---

#### b. $(5x + 1)(3x - 2)$

Use FOIL (First, Outer, Inner, Last):

- First: $5x \cdot 3x = 15x^2$
- Outer: $5x \cdot (-2) = -10x$
- Inner: $1 \cdot 3x = 3x$
- Last: $1 \cdot (-2) = -2$

Combine: $15x^2 - 10x + 3x - 2 = 15x^2 - 7x - 2$

✔ Final Answer:
$$
\boxed{15x^2 - 7x - 2}
$$

> ✔️ Correct!

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#### c. $(x + 4)(x^2 - 2x + 1)$

Use distributive property (multiply each term):

- $x(x^2 - 2x + 1) = x^3 - 2x^2 + x$
- $4(x^2 - 2x + 1) = 4x^2 - 8x + 4$

Add them:
- $x^3$
- $-2x^2 + 4x^2 = 2x^2$
- $x - 8x = -7x$
- $+4$

✔ Final Answer:
$$
\boxed{x^3 + 2x^2 - 7x + 4}
$$

> ✔️ Correct!

---

#### d. $(3x - 4)(x^2 + 4x - 5)$

Distribute:

- $3x(x^2 + 4x - 5) = 3x^3 + 12x^2 - 15x$
- $-4(x^2 + 4x - 5) = -4x^2 - 16x + 20$

Add:
- $3x^3$
- $12x^2 - 4x^2 = 8x^2$
- $-15x - 16x = -31x$
- $+20$

✔ Final Answer:
$$
\boxed{3x^3 + 8x^2 - 31x + 20}
$$

> ✔️ Correct!

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Right Side Problems (More Multiplication)

#### a. $(3x^2 + 5x - 1)(2x^2 + x - 4)$

We’ll multiply term-by-term:

1. $3x^2(2x^2 + x - 4) = 6x^4 + 3x^3 - 12x^2$
2. $5x(2x^2 + x - 4) = 10x^3 + 5x^2 - 20x$
3. $-1(2x^2 + x - 4) = -2x^2 - x + 4$

Now add all together:

- $6x^4$
- $3x^3 + 10x^3 = 13x^3$
- $-12x^2 + 5x^2 - 2x^2 = -9x^2$
- $-20x - x = -21x$
- $+4$

✔ Final Answer:
$$
\boxed{6x^4 + 13x^3 - 9x^2 - 21x + 4}
$$

> ✔️ Your answer is correct!

---

#### b. $(x + 2)(x - 3)(x - 5)$

Do two at a time.

First: $(x + 2)(x - 3)$

- $x(x - 3) = x^2 - 3x$
- $2(x - 3) = 2x - 6$
- Total: $x^2 - x - 6$

Now multiply: $(x^2 - x - 6)(x - 5)$

- $x^2(x - 5) = x^3 - 5x^2$
- $-x(x - 5) = -x^2 + 5x$
- $-6(x - 5) = -6x + 30$

Add:
- $x^3$
- $-5x^2 - x^2 = -6x^2$
- $5x - 6x = -x$
- $+30$

✔ Final Answer:
$$
\boxed{x^3 - 6x^2 - x + 30}
$$

> ✔️ Correct!

---

#### c. $(2x + 1)(2x - 1)$

This is a difference of squares:
$$
(a + b)(a - b) = a^2 - b^2
$$

So:
$$
(2x)^2 - (1)^2 = 4x^2 - 1
$$

✔ Final Answer:
$$
\boxed{4x^2 - 1}
$$

> ✔️ Correct!

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#### d. $(4x + 3)^2$

Square the binomial:
$$
(a + b)^2 = a^2 + 2ab + b^2
$$

- $a = 4x$, $b = 3$
- $(4x)^2 = 16x^2$
- $2(4x)(3) = 24x$
- $3^2 = 9$

✔ Final Answer:
$$
\boxed{16x^2 + 24x + 9}
$$

> ✔️ Correct!

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#### e. $(2x - 3)^2$

Again, use square formula:
- $(2x)^2 = 4x^2$
- $2(2x)(-3) = -12x$
- $(-3)^2 = 9$

✔ Final Answer:
$$
\boxed{4x^2 - 12x + 9}
$$

> ✔️ Correct!

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Summary of Corrections & Feedback

✔ All your answers are correct except:

- Example 4 (Vertical Subtraction): You wrote $+14$ instead of $-4$ for the constant term.
- ✔ Corrected: $\boxed{-2x^4 - 2x^3 - 3x^2 + 6x - 4}$

All other work is accurate and well-documented!

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