SOLUTION: Algebra 2 adding subtracting multiplying polynomials ... - Free Printable

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!

---

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!

---

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!

---

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 do this carefully:

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

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

> ✔️ Your written work shows this correctly.

---

Example 5 – Multiply

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

Wait! There's a typo: both are $3x^2$. Let’s assume it's meant to be:

$$
4x^2(3x^3 + 3x^2 - 2x - 4)
$$

But based on your handwriting, you wrote:
> $4x^2(3x^2 + 3x^2 - 2x - 4)$ → That would be $4x^2(6x^2 - 2x - 4)$

But that doesn’t make sense. Probably a typo in the problem.

Looking at your solution:
You wrote:
$$
12x^4 + 12x^3 - 8x^3 - 16x^2
$$

Wait — that suggests the original was:

$$
4x^2(3x^2 + 3x - 2x - 4) \quad \text{or} \quad 4x^2(3x^2 + x - 4)
$$

But you have:
$$
4x^2(3x^2 + 3x^2 - 2x - 4) = 4x^2(6x^2 - 2x - 4)
$$

Then:
- $4x^2 \cdot 6x^2 = 24x^4$
- $4x^2 \cdot (-2x) = -8x^3$
- $4x^2 \cdot (-4) = -16x^2$

So:
$$
\boxed{24x^4 - 8x^3 - 16x^2}
$$

But your answer says $12x^4 + 12x^3 - 8x^3 - 16x^2$, which simplifies to $12x^4 + 4x^3 - 16x^2$

That suggests the original might be:

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

Then:
- $4x^2 \cdot 3x^2 = 12x^4$
- $4x^2 \cdot x = 4x^3$
- $4x^2 \cdot (-4) = -16x^2$

So: $12x^4 + 4x^3 - 16x^2$

But you wrote $12x^4 + 12x^3 - 8x^3 - 16x^2$, which implies $+12x^3$ from somewhere.

Wait — maybe the original is:

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

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

So:
$$
12x^4 + (12x^3 - 8x^3) - 16x^2 = 12x^4 + 4x^3 - 16x^2
$$

But you wrote: $12x^4 + 12x^3 - 8x^3 - 16x^2$, which is correct only if you're distributing separately.

So yes, your steps are correct, but final simplification should be:

$$
\boxed{12x^4 + 4x^3 - 16x^2}
$$

> ✘ You wrote $12x^4 + 12x^3 - 8x^3 - 16x^2$, which is not simplified.

✔ Corrected: $\boxed{12x^4 + 4x^3 - 16x^2}$

---

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

Use FOIL:
- First: $3x \cdot 3x = 9x^2$
- Outer: $3x \cdot (-2) = -6x$
- Inner: $1 \cdot 3x = 3x$
- Last: $1 \cdot (-2) = -2$

Add:
- $9x^2$
- $-6x + 3x = -3x$
- $-2$

✔ Final Answer: $\boxed{9x^2 - 3x - 2}$

✔️ Your answer is correct!

---

#### c. $(x + 8)(x^2 - 2x + 1)$

Use distributive property (multiply each term):

$$
x(x^2 - 2x + 1) = x^3 - 2x^2 + x
$$
$$
8(x^2 - 2x + 1) = 8x^2 - 16x + 8
$$

Add:
- $x^3$
- $-2x^2 + 8x^2 = 6x^2$
- $x - 16x = -15x$
- $+8$

✔ Final Answer: $\boxed{x^3 + 6x^2 - 15x + 8}$

✔️ Your answer is correct!

---

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

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

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

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

✔️ Your answer is correct!

---

Right Side – More Multiplication

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

This is a full multiplication of two trinomials.

We’ll use distributive method:

First, distribute each term in the first polynomial:

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 matches!

---

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

First multiply two binomials:

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

Now multiply by $(x - 5)$:

$$
(x^2 - x - 6)(x - 5)
$$

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

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

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

✔️ Your answer is correct!

---

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

This is a difference of squares:

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

✔ Final Answer: $\boxed{4x^2 - 1}$

✔️ Correct!

---

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

Square the binomial:
$$
(4x + 3)^2 = (4x)^2 + 2(4x)(3) + (3)^2 = 16x^2 + 24x + 9
$$

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

✔️ Correct!

---

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

$$
(2x)^2 - 2(2x)(3) + (3)^2 = 4x^2 - 12x + 9
$$

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

✔️ Correct!

---

✔ Summary of Corrections & Confirmations:

| Problem | Your Work | Correct? | Correction |
|--------|-----------|---------|------------|
| Example 1 | $5x^3 - 2x^2 - 5x - 17$ | ✔ Yes | None |
| Example 2 | $8x^3 + 5x^2 + 4x + 3$ | ✔ Yes | None |
| Example 3 | $-4x^4 - 5x^3 - 7x^2 - x + 6$ | ✔ Yes | None |
| Example 4 | $-2x^4 - 2x^3 - 3x^2 + 6x - 4$ | ✔ Yes | None |
| Example 5a | $12x^4 + 12x^3 - 8x^3 - 16x^2$ | ⚠️ Partial | Simplify to $12x^4 + 4x^3 - 16x^2$ |
| Others | All correct | ✔ | None |

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✔ Final Notes:

- You did an excellent job solving most problems.
- The only issue is not simplifying fully in Example 5a — combine like terms!
- Otherwise, all steps and answers are correct.

Let me know if you'd like a cleaned-up version of these notes!