Match Binomial Products: (2x+y)(x+2y) and Related Expressions

Q: Why do I get four terms after multiplying binomials?

When you multiply two binomials, each term in the first parentheses multiplies each term in the second. That's 2 × 2 = 4 terms before combining like terms!

Q: How do I handle negative signs in binomials?

Treat the negative sign as part of the term. For $ (2x-y) $, the second term is -y. When multiplying: $ (-y) \times (-2y) = +2y^2 $.

Q: What if I can't find matching expressions in the answer choices?

Double-check your like term combination! The most common error is forgetting to add xy terms together. Make sure 4xy + xy = 5xy, not leaving them separate.

Q: Is there a pattern to remember for FOIL?

FOIL helps: First, Outer, Inner, Last. For $ (2x+y)(x+2y) $: First = 2x·x, Outer = 2x·2y, Inner = y·x, Last = y·2y.

Q: Why do some expressions have +5xy and others -5xy?

The middle coefficient depends on signs. When both binomials have same signs $ (+)(+) $, you get positive. When signs differ $ (+)(-) $, you get negative results!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Open parentheses

00:03 Open parentheses properly, multiply each factor by each factor

00:15 Calculate the products

00:22 Collect terms

00:27 This is the simplification for 1, let's continue to 2

00:33 Open parentheses properly, multiply each factor by each factor

00:40 Calculate the products

00:45 Collect terms

00:51 This is the simplification for 2, let's continue to 3

00:57 Open parentheses properly, multiply each factor by each factor

01:05 Calculate the products

01:14 Collect terms

01:18 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Match the expressions that have the same value:

$(2x+y)(x+2y)$
$(2x+2y)(x+y)$
$(2x-y)(x-2y)$
a. $2x^2+4xy+2y^2$
b. $2x^2-5xy+2y^2$
c. $2x^2+5xy+2y^2$

2

Step-by-step solution

Let's simplify the given expressions, open the parentheses using the expanded distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$ In the formula template for the above distribution law, we take by default that the operation between the terms inside of the parentheses is addition. Note that the sign preceding the term is an inseparable part of it. Furthermore we will apply the laws of sign multiplication to our expression. We will then open the parentheses using the above formula, where there is an addition operation between all terms.

Proceed to simplify each of the expressions in the given problem, whilst making sure to open the parentheses using the mentioned distribution law, the commutative law of addition and multiplication and combining like terms (if there are like terms in the expression obtained after opening the parentheses):

$(2x+y)(x+2y) \\ 2x\cdot x+2x\cdot 2y+y\cdot x+y\cdot2y\\ 2x^2+4xy+xy+2y^2\\ \boxed{2x^2+5xy+2y^2}$
$(2x+2y)(x+y) \\ 2x\cdot x+2x\cdot y+2y\cdot x+2y\cdot y\\ 2x^2+2xy+2yx+2y^2\\ \boxed{2x^2+4xy+2y^2}$
$(2x-y)(x-2y) \\ \downarrow\\ \big(2x+(-y)\big)\big(x+(-2y)\big) \\ 2x\cdot x+2x\cdot(-2y)+(-y)\cdot x+(-y)\cdot(-2y)\\ 2x^2-4xy-yx+2y^2\\ \boxed{2x^2-5xy+2y^2}$
Now, we'll use the commutative law of addition and multiplication to notice that:
The simplified expression in 1 matches the expression in option C,
The simplified expression in 2 matches the expression in option A,
The simplified expression in 3 matches the expression in option B,

Therefore, the correct answer (among the suggested options) is answer A.

3

Final Answer

1-c, 2-a, 3-b

Key Points to Remember

Essential concepts to master this topic

Distribution: Each term multiplies every term in the other parentheses
Technique: $(2x+y)(x+2y) = 2x^2+4xy+xy+2y^2$ then combine
Check: Count xy terms carefully: 4xy + xy = 5xy ✓

Common Mistakes

Avoid these frequent errors

Forgetting to combine like terms after distribution
Don't leave $2x^2+4xy+xy+2y^2$ as final answer = missing the combination step! This gives incomplete expressions that don't match the answer choices. Always combine like terms: 4xy + xy = 5xy.

Practice Quiz

Test your knowledge with interactive questions

$$ (x+y)(x-y)= $$

FAQ

Everything you need to know about this question

Why do I get four terms after multiplying binomials?

+

When you multiply two binomials, each term in the first parentheses multiplies each term in the second. That's 2 × 2 = 4 terms before combining like terms!

How do I handle negative signs in binomials?

+

Treat the negative sign as part of the term. For $(2x-y)$ , the second term is -y. When multiplying: $(-y) \times (-2y) = +2y^2$ .

What if I can't find matching expressions in the answer choices?

+

Double-check your like term combination! The most common error is forgetting to add xy terms together. Make sure 4xy + xy = 5xy, not leaving them separate.

Is there a pattern to remember for FOIL?

+

FOIL helps: First, Outer, Inner, Last. For $(2x+y)(x+2y)$ : First = 2x·x, Outer = 2x·2y, Inner = y·x, Last = y·2y.

Why do some expressions have +5xy and others -5xy?

+

The middle coefficient depends on signs. When both binomials have same signs $(+)(+)$ , you get positive. When signs differ $(+)(-)$ , you get negative results!

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