Match Equivalent Expressions: (4+x)(y+8+x) and Related Terms

Q: How do I keep track of all the terms when expanding?

Use the distributive property systematically! For $ (4+x)(y+8+x) $, first multiply 4 by each term: 4y + 32 + 4x, then multiply x by each term: xy + 8x + x².

Q: Why do some terms combine and others don't?

Like terms have the same variables with the same powers. For example, 4x + 8x = 12x, but xy and x² are different because they have different variable combinations.

Q: What's the best way to organize my work?

Write each multiplication step clearly, then group like terms together. Use colors or underlining to identify terms that can be combined: all x terms, all y terms, all x² terms, etc.

Q: How can I check if my expansion is correct?

Substitute simple values like x = 1, y = 1 into both the original expression and your expanded form. If they give the same result, your expansion is likely correct!

Q: What if I get confused with the order of terms?

The order doesn't matter for the final answer! $ x^2 + 12x + xy + 4y + 32 $ is the same as $ 4y + x^2 + 32 + 12x + xy $. Focus on having all the right terms with correct coefficients.

Polynomial Expansion with Multiple Variables

Join expressions of equal value

$(4+x)(y+8+x)$
$(4+x+y)(8+x)$
$(12+x)(y+x)$
a. $x^2+12x+xy+12y$
b. $x^2+12x+xy+4y+32$
c. $x^2+12x+xy+8y+32$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:28 First, let's open the parentheses.

00:31 Now, carefully multiply each factor by every other factor. Take your time.

00:47 Great job! Next, calculate the products and gather similar terms together.

00:55 Well done on simplifying the first part. Let's move on to part two.

01:00 Again, open the parentheses and multiply each factor with each other. You're doing great!

01:11 Once more, calculate the products and group like terms.

01:16 Fantastic! That's the second simplification. Let's go to part three.

01:22 Open the parentheses. Keep multiplying each factor with the others.

01:29 Finally, find the products and combine similar terms together.

01:34 Awesome! You've reached the solution. And that's how we solve this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Join expressions of equal value

$(4+x)(y+8+x)$
$(4+x+y)(8+x)$
$(12+x)(y+x)$
a. $x^2+12x+xy+12y$
b. $x^2+12x+xy+4y+32$
c. $x^2+12x+xy+8y+32$

Step-by-step solution

To solve this problem, we'll expand each expression and find which polynomial they correspond with:

Start with expression 1: $(4+x)(y+8+x)$ .

Using the distributive property, expand as follows:
$= 4(y+8+x) + x(y+8+x)$
$= 4y + 32 + 4x + xy + 8x + x^2$
$= x^2 + xy + 12x + 4y + 32$

This matches the polynomial $x^2 + 12x + xy + 4y + 32$ , which is option b.

Next, consider expression 2: $(4+x+y)(8+x)$ .

Once again, expand using distributive property:
$= (4+x+y)8 + (4+x+y)x$
$= 32 + 8x + 8y + 4x + x^2 + xy$
$= x^2 + 12x + xy + 8y + 32$

This matches the polynomial $x^2 + 12x + xy + 8y + 32$ , which is option c.

Finally, consider expression 3: $(12+x)(y+x)$ .

Use distributive property to expand:
$= 12(y+x) + x(y+x)$
$= 12y + 12x + xy + x^2$
$= x^2 + 12x + xy + 12y$

This matches the polynomial $x^2 + 12x + xy + 12y$ , which is option a.

The correct matches are therefore: 1-c, 2-b, 3-a.

Final Answer

1-c, 2-b, 3-a

Key Points to Remember

Essential concepts to master this topic

FOIL Method: Distribute each term to all terms in second expression
Technique: For (4+x)(y+8+x), multiply 4 by each term, then x by each term
Check: Count terms and verify coefficients match the given options ✓

Common Mistakes

Avoid these frequent errors

Missing terms when expanding
Don't skip terms during distribution = incomplete polynomial! Students often forget to multiply every term in the first expression by every term in the second. Always ensure each term from the first expression multiplies with each term from the second expression.

Practice Quiz

Test your knowledge with interactive questions

$(3+20)\times(12+4)=$

FAQ

Everything you need to know about this question

How do I keep track of all the terms when expanding?

Use the distributive property systematically! For $(4+x)(y+8+x)$ , first multiply 4 by each term: 4y + 32 + 4x, then multiply x by each term: xy + 8x + x².

Why do some terms combine and others don't?

Like terms have the same variables with the same powers. For example, 4x + 8x = 12x, but xy and x² are different because they have different variable combinations.

What's the best way to organize my work?

Write each multiplication step clearly, then group like terms together. Use colors or underlining to identify terms that can be combined: all x terms, all y terms, all x² terms, etc.

How can I check if my expansion is correct?

Substitute simple values like x = 1, y = 1 into both the original expression and your expanded form. If they give the same result, your expansion is likely correct!

What if I get confused with the order of terms?

The order doesn't matter for the final answer! $x^2 + 12x + xy + 4y + 32$ is the same as $4y + x^2 + 32 + 12x + xy$ . Focus on having all the right terms with correct coefficients.

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