Adding and subtracting polynomials: Fill out & sign online | DocHub - Free Printable

Let's solve each polynomial addition and subtraction problem step by step, simplify the expressions, and then match the answers to the corresponding letters in the alphabet grid.

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Problem 1:

(2x² + 4x + 1) + (3x² + 6x + 5)
Add like terms:
- $2x^2 + 3x^2 = 5x^2$
- $4x + 6x = 10x$
- $1 + 5 = 6$
✔ Answer: $5x^2 + 10x + 6$ → This matches U

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Problem 2:

(x + 6) + (3x + 5)
- $x + 3x = 4x$
- $6 + 5 = 11$
✔ Answer: $4x + 11$ → This matches B

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Problem 3:

(x³ + 2x² - 4) + (3x² + 6)
- $x^3$ stays
- $2x^2 + 3x^2 = 5x^2$
- $-4 + 6 = 2$
✔ Answer: $x^3 + 5x^2 + 2$ → This matches Y

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Problem 4:

(4x - 2y) + (-2x + 6y)
- $4x - 2x = 2x$
- $-2y + 6y = 4y$
✔ Answer: $2x + 4y$ → This matches O

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Problem 5:

(x² + 6x - 4) + (-x² - 6x + 4)
- $x^2 - x^2 = 0$
- $6x - 6x = 0$
- $-4 + 4 = 0$
✔ Answer: $0$ → This matches E

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Problem 6:

6x - 4 - 6x
- $6x - 6x = 0$
- $-4$ remains
✔ Answer: $-4$ → This matches C

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Problem 7:

3x + 6y - 8y + 2x
- $3x + 2x = 5x$
- $6y - 8y = -2y$
✔ Answer: $5x - 2y$ → This matches A

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Problem 8:

(6x² + 3x - 5) - (4x² - 2x + 3)
Distribute the minus sign:
= $6x^2 + 3x - 5 - 4x^2 + 2x - 3$
Now combine:
- $6x^2 - 4x^2 = 2x^2$
- $3x + 2x = 5x$
- $-5 - 3 = -8$
✔ Answer: $2x^2 + 5x - 8$ → This matches R

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Problem 9:

x² + 6 - 6 + x²
- $x^2 + x^2 = 2x^2$
- $6 - 6 = 0$
✔ Answer: $2x^2$ → This matches S

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Problem 10:

(3m - 6n) + (2m + n)
- $3m + 2m = 5m$
- $-6n + n = -5n$
✔ Answer: $5m - 5n$ → This matches M

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Problem 11:

(5x - 6) - (8x + 2)
Distribute the minus:
= $5x - 6 - 8x - 2$
- $5x - 8x = -3x$
- $-6 - 2 = -8$
✔ Answer: $-3x - 8$ → This matches I

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Problem 12:

(3x² + x - 4) + (-8x² - 4x + 6)
- $3x^2 - 8x^2 = -5x^2$
- $x - 4x = -3x$
- $-4 + 6 = 2$
✔ Answer: $-5x^2 - 3x + 2$ → This matches D

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Problem 13:

(x² + 6x + 5) + (x² + 6x + 5)
- $x^2 + x^2 = 2x^2$
- $6x + 6x = 12x$
- $5 + 5 = 10$
✔ Answer: $2x^2 + 12x + 10$ → This matches K

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Problem 14:

3x(x² + 2x - 6) + 4(x² - 6x + 2)
First distribute:
- $3x \cdot x^2 = 3x^3$
- $3x \cdot 2x = 6x^2$
- $3x \cdot (-6) = -18x$
→ $3x^3 + 6x^2 - 18x$

Second:
- $4 \cdot x^2 = 4x^2$
- $4 \cdot (-6x) = -24x$
- $4 \cdot 2 = 8$
→ $4x^2 - 24x + 8$

Now add both:
- $3x^3$
- $6x^2 + 4x^2 = 10x^2$
- $-18x - 24x = -42x$
- $+8$
✔ Answer: $3x^3 + 10x^2 - 42x + 8$ → This matches N

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Problem 15:

(3x + 6) - (3x - 6)
Distribute the minus:
= $3x + 6 - 3x + 6$
- $3x - 3x = 0$
- $6 + 6 = 12$
✔ Answer: $12$ → This matches T

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Problem 16:

(7x² + 8x - 4) - (6x² + 8x - 6)
Distribute the minus:
= $7x^2 + 8x - 4 - 6x^2 - 8x + 6$
- $7x^2 - 6x^2 = x^2$
- $8x - 8x = 0$
- $-4 + 6 = 2$
✔ Answer: $x^2 + 2$ → This matches L

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## ✔ Final Answers:

| Problem | Answer | Letter |
|--------|--------|--------|
| 1 | $5x^2 + 10x + 6$ | U |
| 2 | $4x + 11$ | B |
| 3 | $x^3 + 5x^2 + 2$ | Y |
| 4 | $2x + 4y$ | O |
| 5 | $0$ | E |
| 6 | $-4$ | C |
| 7 | $5x - 2y$ | A |
| 8 | $2x^2 + 5x - 8$ | R |
| 9 | $2x^2$ | S |
| 10 | $5m - 5n$ | M |
| 11 | $-3x - 8$ | I |
| 12 | $-5x^2 - 3x + 2$ | D |
| 13 | $2x^2 + 12x + 10$ | K |
| 14 | $3x^3 + 10x^2 - 42x + 8$ | N |
| 15 | $12$ | T |
| 16 | $x^2 + 2$ | L |

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Now, fill in the puzzle using the numbers at the top:

The numbers at the top are:

```
7 12 12 7 14 12 9 1 2 15 8 7 6 15
16 11 13 5 15 5 8 10 9 4 14 16 3
```

We now go through each number and find the letter from the answer above.

Let’s do it step by step:

| Number | Problem | Letter |
|--------|--------|--------|
| 7 | Problem 7 → A | A |
| 12 | Problem 12 → D | D |
| 12 | Problem 12 → D | D |
| 7 | Problem 7 → A | A |
| 14 | Problem 14 → N | N |
| 12 | Problem 12 → D | D |
| 9 | Problem 9 → S | S |
| 1 | Problem 1 → U | U |
| 2 | Problem 2 → B | B |
| 15 | Problem 15 → T | T |
| 8 | Problem 8 → R | R |
| 7 | Problem 7 → A | A |
| 6 | Problem 6 → C | C |
| 15 | Problem 15 → T | T |
| 16 | Problem 16 → L | L |
| 11 | Problem 11 → I | I |
| 13 | Problem 13 → K | K |
| 5 | Problem 5 → E | E |
| 15 | Problem 15 → T | T |
| 5 | Problem 5 → E | E |
| 8 | Problem 8 → R | R |
| 10 | Problem 10 → M | M |
| 9 | Problem 9 → S | S |
| 4 | Problem 4 → O | O |
| 14 | Problem 14 → N | N |
| 16 | Problem 16 → L | L |
| 3 | Problem 3 → Y | Y |

Now write them in order:

Row 1:
7, 12, 12, 7, 14, 12, 9, 1, 2, 15, 8, 7, 6, 15
→ A, D, D, A, N, D, S, U, B, T, R, A, C, T

Row 2:
16, 11, 13, 5, 15, 5, 8, 10, 9, 4, 14, 16, 3
→ L, I, K, E, T, E, R, M, S, O, N, L, Y

So the final puzzle is:

```
A D D A N D S U B T R A C T
L I K E T E R M S O N L Y
```

That spells:

> ADD AND SUBTRACT LIKE TERMS ONLY

Which is a quote about combining like terms — perfect for a "quotable puzzle"!

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✔ Final Answer:

The completed puzzle reads:

ADD AND SUBTRACT LIKE TERMS ONLY

Each problem was solved correctly, and the matching letters form this meaningful sentence.