Expand (x+y)(3x+2y): Step-by-Step Binomial Multiplication

Q: What's the easiest way to remember all the multiplications?

Use the FOIL method: First terms (x·3x), Outer terms (x·2y), Inner terms (y·3x), Last terms (y·2y). This ensures you don't miss any combinations!

Q: Why do I get 5xy and not 6xy in the middle term?

You get $ 2xy + 3xy = 5xy $, not 6xy! The outer multiplication gives $ x \cdot 2y = 2xy $ and inner gives $ y \cdot 3x = 3xy $. Add these: 2xy + 3xy = 5xy.

Q: How do I know which terms are 'like terms'?

Like terms have exactly the same variables with the same exponents. For example: $ 2xy $ and $ 3xy $ are like terms, but $ 3x^2 $ and $ 2y^2 $ are not.

Q: Can I use this method for any two binomials?

Yes! The distribution method (or FOIL) works for any two binomials, whether they have positive or negative terms. Just remember to keep track of signs when multiplying.

Q: What if I expand and get a different order of terms?

That's perfectly fine! $ 3x^2 + 5xy + 2y^2 $ equals $ 2y^2 + 5xy + 3x^2 $. The standard form arranges terms from highest to lowest degree, but any correct order works.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solution

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

00:23 Calculate the multiplications

00:41 Collect terms

00:47 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

Expand the following expression:

$(x+y)(3x+2y)=$

2

Step-by-step solution

Let's simplify the given expression by opening the parentheses using the extended distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that 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. We must remember that the sign preceding the term is an inseparable part of it. We will also apply the rules of sign multiplication in order to present any expression in the parentheses, which we will open using the above formula. First as an expression where addition operation exists between all terms. In this expression it's clear that all terms have a plus sign prefix, therefore we'll proceed directly to opening the parentheses,

Let's begin to open the parentheses:

$(\textcolor{red}{x}+\textcolor{blue}{y})(3x+2y)\\ \textcolor{red}{x}\cdot 3x+\textcolor{red}{x}\cdot2y+\textcolor{blue}{y}\cdot 3x+\textcolor{blue}{y} \cdot2y\\ 3x^2+2xy+3xy+2y^2$

In calculating the above multiplications, we used the multiplication table as well as the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step we'll combine like terms, which we'll define as terms where the variable (or variables, each separately), in this case x and y, have identical exponents. (In the absence of one of the variables from the expression, we'll treat its exponent as zero power. This is due to the fact that raising any number to the zero power yields the result 1) We'll apply the commutative law of addition, and proceed to arrange the expression (if needed) from highest to lowest power from left to right (we'll treat the free number as zero power):
$\textcolor{purple}{ 3x^2}\textcolor{green}{+2xy}\textcolor{green}{+3xy}\textcolor{orange}{+2y^2}\\ \textcolor{purple}{ 3x^2}\textcolor{green}{+5xy}\textcolor{orange}{+2y^2}\\$ We highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term remains an inseparable part of it,

We therefore concluded that the correct answer is answer A.

3

Final Answer

$3x^2+5xy+2y^2$

Key Points to Remember

Essential concepts to master this topic

FOIL Method: First, Outer, Inner, Last terms multiply systematically
Distribution: Each term in first binomial times each in second: $x \cdot 3x = 3x^2$
Combine Like Terms: Add coefficients of same variables: $2xy + 3xy = 5xy$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all four term combinations
Don't just multiply x·3x and y·2y = 3x² + 2y²! This misses the crucial cross terms. You get 3x² + 2y² instead of 3x² + 5xy + 2y². Always multiply every term in the first binomial by every term in the second binomial.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

What's the easiest way to remember all the multiplications?

+

Use the FOIL method: First terms (x·3x), Outer terms (x·2y), Inner terms (y·3x), Last terms (y·2y). This ensures you don't miss any combinations!

Why do I get 5xy and not 6xy in the middle term?

+

You get $2xy + 3xy = 5xy$ , not 6xy! The outer multiplication gives $x \cdot 2y = 2xy$ and inner gives $y \cdot 3x = 3xy$ . Add these: 2xy + 3xy = 5xy.

How do I know which terms are 'like terms'?

+

Like terms have exactly the same variables with the same exponents. For example: $2xy$ and $3xy$ are like terms, but $3x^2$ and $2y^2$ are not.

Can I use this method for any two binomials?

+

Yes! The distribution method (or FOIL) works for any two binomials, whether they have positive or negative terms. Just remember to keep track of signs when multiplying.

What if I expand and get a different order of terms?

+

That's perfectly fine! $3x^2 + 5xy + 2y^2$ equals $2y^2 + 5xy + 3x^2$ . The standard form arranges terms from highest to lowest degree, but any correct order works.

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Expand (x+y)(3x+2y): Step-by-Step Binomial Multiplication

Binomial Expansion with Distribution Method

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