GCF of Polynomials: all variables Worksheet for 7th - 12th Grade ... - Free Printable

Let’s go step by step to fill in the right column — we’re matching expressions that are equal (same value) or showing how they relate.

We’ll use basic algebra rules:
- Commutative property: a + b = b + a, and ab = ba
- Distributive property: a(b + c) = ab + ac
- Squaring binomials: (a + b)² = a² + 2ab + b²; (a – b)² = a² – 2ab + b²
- Difference of squares: (a – b)(a + b) = a² – b²

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Row 1:
Left: `(a + b)` and `(a + b)²`
→ These are NOT equal. One is linear, one is squared.
But maybe the task wants us to write what each equals? Or match with equivalent forms?

Wait — looking at other rows, it seems like for each row, we have two expressions on the left, and we need to say whether they are equal, or show their relationship.

Actually, re-examining the table structure:

Each row has two expressions on the left, and an empty box on the right. The title says “Answers”, so likely we are to write if they are equal, or simplify/show equivalence.

But look at Row 3:
`(x - y)` and `(-x - y)` → these are not equal. But note: (-x - y) = -(x + y), while (x - y) is different.

Wait — perhaps the instruction is to determine if the two expressions in each row are equal? Let’s test that.

Row 1: (a + b) vs (a + b)² → Not equal unless a+b=0 or 1. So generally, no.

But then why would they be paired? Maybe it's about expanding or rewriting?

Look at Row 5:
`(a + b)c` and `c(a + b)` → These ARE equal because multiplication is commutative: c times (a+b) is same as (a+b) times c.

Similarly, Row 6: `(a + b)(c + d)` and `(a - b)(c + d)` → Not equal, unless b=0.

Hmm.

Another idea: Perhaps the right column is for writing the simplified form or expanded form of the first expression? But then why two expressions per row?

Wait — let’s read the pairs again carefully.

Actually, I think the task is: For each pair of expressions on the left, determine if they are equal. If yes, write “equal” or just leave blank? But some rows already have answers filled in on the left? No — looking back, the user provided an image with only the left side filled, and right side blank. And the header says “Answers”, so probably we are to complete the right side with the correct relationship or simplification.

But in Row 4:
`m² + n²` and `(m + n)` → clearly not equal. But wait, maybe it’s a typo? Should it be `(m + n)²`? Because m² + n² ≠ (m + n)².

Unless... perhaps the second expression is meant to be compared after expansion?

I think there might be a misinterpretation.

Alternative approach: Look at Row 7:
`(a + b)c` and `ac + bc` → This is distributive property! So (a + b)c = ac + bc. So they are equal.

Similarly, Row 8: `(a + b)(c + d)` and `(a - b)(c + d)` → not equal.

But Row 9: `(a + b)(c - d)` and `(a - b)(c + d)` → also not equal.

Row 10: `xy(x + y)` and `xy(x - y)` → not equal.

Row 11: `xy` and `yx` → equal, since multiplication is commutative.

Ah! Now I see the pattern.

In many rows, the two expressions are either equal due to properties (commutative, distributive), or they are different.

But in Row 1: (a + b) and (a + b)² — definitely not equal.

Unless... perhaps the task is to expand the second expression and compare? But that doesn’t make sense for all rows.

Wait — let’s list all rows with my analysis:

Row 1: (a + b) and (a + b)² → Not equal. Right side should indicate they are not equal? But how?

Perhaps the right column is for writing the result of operations or equivalences.

Another thought: Maybe for each row, the first expression is to be rewritten or matched with the second using algebraic identities.

Let me try to interpret based on common textbook exercises.

Often, such tables ask: "Are the following expressions equal? Write Yes or No."

But in Row 5: (a + b)c and c(a + b) → Yes, equal.

Row 11: xy and yx → Yes.

Row 7: (a + b)c and ac + bc → Yes, by distributive property.

Row 3: (x - y) and (-x - y) → No, because -x - y = -(x + y), which is not the same as x - y.

Row 4: m² + n² and (m + n) → No, and also (m + n) is not even squared — probably a typo, but as written, no.

Row 6: (a + b)(c + d) and (a - b)(c + d) → Only equal if b=0, so generally no.

Row 8: (a + b)(c + d) and (a - b)(c + d) — same as row 6? Wait no, row 6 is (a + b)(c + d) and (a - b)(c + d)? In the original, row 6 is: (a + b)(c + d) and (a - b)(c + d)? Let me check the user's input.

From user's text:

Row 6: (a + b)(c + d) and (a - b)(c + d) — yes.

But in row 8: (a + b)(c + d) and (a - b)(c + d) — wait, no, in the user's message, row 8 is: (a + b)(c + d) and (a - b)(c + d)? Let's copy exactly:

User wrote:

Row 6: (a + b)(c + d) and (a - b)(c + d)

Row 8: (a + b)(c + d) and (a - b)(c + d) — wait, no, in the initial post, it's:

Looking back:

In the user's message, the rows are:

After "Answers:"

First row: (a + b) and (a + b)^2

Second: (x - y) and (-x - y)

Third: (x - y) and (-x - y) — wait, duplicate? No, let's list them as given:

The user pasted:

(a + b) and (a + b)^2 |

(x - y) and (-x - y) |

(m^2 + n^2) and (m + n) |

(a + b)c and c(a + b) |

(a + b)c and ac + bc |

(a + b)(c + d) and (a - b)(c + d) |

(a + b)(c + d) and (a - b)(c + d) — wait, this is repeated? No, in the text:

Actually, from the user's input:

" (a + b) and (a + b)^2 |

(x - y) and (-x - y) |

m^2 + n^2 and (m + n) |

(a + b)c and c(a + b) |

(a + b)c and ac + bc |

(a + b)(c + d) and (a - b)(c + d) |

(a + b)(c + d) and (a - b)(c + d) " — wait, that can't be, probably a copy error.

Let me read the user's message carefully:

They wrote:

"(a + b) and (a + b)^2 |

(x - y) and (-x - y) |

m^2 + n^2 and (m + n) |

(a + b)c and c(a + b) |

(a + b)c and ac + bc |

(a + b)(c + d) and (a - b)(c + d) |

(a + b)(c + d) and (a - b)(c + d) " — oh, it's listed twice? That must be a mistake.

No, in the original, it's:

From the user:

" (a + b) and (a + b)^2 |

(x - y) and (-x - y) |

m^2 + n^2 and (m + n) |

(a + b)c and c(a + b) |

(a + b)c and ac + bc |

(a + b)(c + d) and (a - b)(c + d) |

(a + b)(c + d) and (a - b)(c + d) " — wait, the last two are identical? That doesn't make sense.

Perhaps it's a formatting issue. Let me count the lines.

In the user's message, after "Answers:", there are 11 rows? Let's list them as numbered:

1. (a + b) and (a + b)^2

2. (x - y) and (-x - y)

3. m^2 + n^2 and (m + n)

4. (a + b)c and c(a + b)

5. (a + b)c and ac + bc

6. (a + b)(c + d) and (a - b)(c + d)

7. (a + b)(c + d) and (a - b)(c + d) — this is duplicate of 6? Probably a typo.

Then:

8. (a + b)(c - d) and (a - b)(c + d)

9. xy(x + y) and xy(x - y)

10. xy and yx

And that's 10 rows? But in the text, it's written as 11 items? Let's see the user's input:

" (a + b) and (a + b)^2 |

(x - y) and (-x - y) |

m^2 + n^2 and (m + n) |

(a + b)c and c(a + b) |

(a + b)c and ac + bc |

(a + b)(c + d) and (a - b)(c + d) |

(a + b)(c + d) and (a - b)(c + d) |

(a + b)(c - d) and (a - b)(c + d) |

xy(x + y) and xy(x - y) |

xy and yx "

That's 10 rows, but row 6 and 7 are identical. Likely, row 7 is meant to be something else, but as per user, it's repeated.

Perhaps in the image, it's different, but since we can't see, we'll assume the text is correct.

To resolve, I'll assume that for each pair, we need to state if they are equal or not, or provide the correct relationship.

But in educational context, often for such tables, the right column is for writing "Equal" or "Not Equal", or sometimes for simplifying.

Given that in row 5: (a + b)c and ac + bc are equal by distributive property.

Row 4: (a + b)c and c(a + b) are equal by commutative property of multiplication.

Row 11: xy and yx are equal.

Row 7: if it's (a + b)(c + d) and (a - b)(c + d), then not equal.

But let's proceed with the most logical interpretation: for each pair, determine if the two expressions are algebraically equal for all values of the variables. If yes, write "Equal"; if not, write "Not Equal".

However, in some cases, like row 1, (a + b) and (a + b)^2 are never equal except for specific values, so generally not equal.

Similarly, row 3: m^2 + n^2 and (m + n) — not equal, and also dimensions don't match.

Row 2: (x - y) and (-x - y) — let's calculate: -x - y = -(x + y), while x - y is different, so not equal.

Row 8: (a + b)(c - d) and (a - b)(c + d) — expand both:

First: a(c - d) + b(c - d) = ac - ad + bc - bd

Second: a(c + d) - b(c + d) = ac + ad - bc - bd

Compare: ac - ad + bc - bd vs ac + ad - bc - bd — not the same, unless ad=0 and bc=0, so generally not equal.

Row 9: xy(x + y) and xy(x - y) = x^2y + xy^2 vs x^2y - xy^2 — not equal.

Row 10: xy and yx — equal.

Now, for row 6 and 7 being the same, perhaps it's a mistake, but we'll treat them separately.

Perhaps the right column is for writing the expanded form or something else.

Another idea: perhaps for each row, the first expression is to be manipulated to match the second, or vice versa.

But that seems complicated.

Let's look for clues in the rows that are obviously equal.

Row 4: (a + b)c and c(a + b) — equal, so right side could be "Equal" or "Commutative Property".

Row 5: (a + b)c and ac + bc — equal, "Distributive Property".

Row 11: xy and yx — "Commutative Property".

For others, "Not Equal".

But in row 1, (a + b) and (a + b)^2 — not equal.

Perhaps the task is to identify which pairs are equivalent.

I recall that in some worksheets, they have pairs and you circle if they are equal.

Given the constraints, I think the safest way is to for each row, write "Equal" if the two expressions are always equal for all real numbers, otherwise "Not Equal".

Let's do that.

Row 1: (a + b) vs (a + b)^2 — Not Equal (unless a+b=0 or 1)

Row 2: (x - y) vs (-x - y) — Not Equal (e.g., x=1,y=0: 1 vs -1)

Row 3: m^2 + n^2 vs (m + n) — Not Equal (e.g., m=1,n=1: 2 vs 2, wait 1+1=2, m^2+n^2=1+1=2, so equal in this case, but if m=2,n=0: 4+0=4, m+n=2, not equal. So not always equal.)

Row 4: (a + b)c vs c(a + b) — Equal (commutative property of multiplication)

Row 5: (a + b)c vs ac + bc — Equal (distributive property)

Row 6: (a + b)(c + d) vs (a - b)(c + d) — Not Equal (e.g., a=1,b=1,c=1,d=1: (2)(2)=4 vs (0)(2)=0)

Row 7: same as row 6 — Not Equal

Row 8: (a + b)(c - d) vs (a - b)(c + d) — Not Equal (as calculated earlier)

Row 9: xy(x + y) vs xy(x - y) — Not Equal (e.g., x=1,y=1: 1*2=2 vs 1*0=0)

Row 10: xy vs yx — Equal

So for the right column, we can write "Equal" or "Not Equal".

But in the table, the right column is empty, so we fill it with that.

Perhaps they want the reason, but the instruction is to solve accurately, and for a student, simple answer.

Since the user said "Solve the problem accurately", and the problem is to complete the table, I'll provide for each row whether the two expressions are equal.

So final decision:

For each row, in the right column, write "Equal" if the two expressions are algebraically identical for all values of the variables, otherwise "Not Equal".

Now, let's list:

1. (a + b) and (a + b)^2 → Not Equal

2. (x - y) and (-x - y) → Not Equal

3. m^2 + n^2 and (m + n) → Not Equal

4. (a + b)c and c(a + b) → Equal

5. (a + b)c and ac + bc → Equal

6. (a + b)(c + d) and (a - b)(c + d) → Not Equal

7. (a + b)(c + d) and (a - b)(c + d) → Not Equal (same as 6)

8. (a + b)(c - d) and (a - b)(c + d) → Not Equal

9. xy(x + y) and xy(x - y) → Not Equal

10. xy and yx → Equal

Note: Row 7 is duplicate of row 6, but we'll keep it as is.

Perhaps in the original image, row 7 is different, but based on user input, we proceed.

So for the Final Answer, since the task is to fill the right column, I'll provide the answers for each row.

But the user didn't specify how to output, but since it's a table, perhaps list them in order.

Since the response format requires Final Answer at the end, and no other sections, I'll write the answers for each row.

To make it clear, I'll number them.

Final Answer:

1. Not Equal
2. Not Equal
3. Not Equal
4. Equal
5. Equal
6. Not Equal
7. Not Equal
8. Not Equal
9. Not Equal
10. Equal

But let's double-check row 3: m^2 + n^2 and (m + n). Is there any chance they are equal? Only if m^2 + n^2 = m + n, which is not true in general. For example, m=2, n=2: left=4+4=8, right=4, not equal. So correct.

Row 2: x=2,y=1: left=2-1=1, right=-2-1=-3, not equal.

Row 8: a=2,b=1,c=3,d=1: left=(3)(2)=6, right=(1)(4)=4, not equal.

All good.

So the completed right column is as above.