homework 3 multiplying binomials and trinomials answers

Multiplying Binomials

Once they achieve mastery in adding and subtracting polynomials, Algebra 1 students go on to learn how to multiply or divide polynomials, with the help of the distributive property. Before learning how to multiply long expressions, students engage in multiplying binomials.

If you’re a math teacher or homeschooling parent teaching multiplying binomials to your students, in this article, we bring you some ideas that will assist you in your effort to make your teaching more effective. Read on to learn more!

How to Teach Multiplying Binomials

What are binomials.

For starters, you can remind children what binomials are. Binomials are a type of polynomials, representing a sum or difference of two terms, that is, of two monomials. These terms can be a variable, a constant, or an exponent.

Provide a few examples, such as:

3 m² – 7

13x – 1

Review the Distributive Property

Since learning how to multiply binomials requires having a solid foundation in the distributive property, it makes sense to review it.

We can present the distributive property of multiplication over addition as:

a x (b + c) = a x b + a x c

(b + c) x a = b x a + c x a

Whereas the distributive property of multiplication over subtraction is presented as:

a x (b – c) = a x b – a x c

(b – c) x a = b x a – c x a

You can also check out our article on teaching the distributive property.

How to Multiply Binomials

There are several methods that students learn to multiply binomials, such as multiplying binomials vertically, multiplying binomials horizontally, or multiplying binomials using a grid. There is also a special method, called the FOIL method, used only for multiplying one binomial by another binomial.

In this article, we’ll focus on the FOIL method, but you can also refer to these Guided Notes (Members Only) for more instructions on the other methods.

In addition, if you have the possibility, you can also choose to show a few videos at the beginning of the lesson or throughout it in order to help children understand the process of multiplication better.

This video by Khan Academy is a free resource on how to multiply binomials by using the grid or area model, and this video contains step-by-step instructions on using the FOIL method.

Multiplying Tiny Expressions

First provide an example of multiplication of a tiny expression, such as monomials:

2x multiplied by 3 = 6x

Just like two apples times three is 6 apples!

2x multiplied by 3x = 6x²

So we multiply the x’s as well!

Multiplying Binomials: Example 1

So far, so good! But if we look at two binomials and try to multiply them, things get a bit more complicated. Provide an example of two such binomials:

(x + 8) (x – 2) = ?

Explain to students that the FOIL method stands for: First, Outer, Inner, and Last. So we have to multiply in this exact order and the abbreviation FOIL helps us remember the order! Remind students to carry the sign.

You can also illustrate the First, Outer, Inner, and Last order on the whiteboard by using markers in different colors so that it’s easier for children to remember it:

In other words, we’ll have:

First: x multiplied by x = x²

Outer: x multiplied by 2 = 2x

Inner: 8 multiplied by x = 8x

Last: 8 multiplied by 2 = 16

By doing this, we are basically distributing each term in one binomial across the other binomial term. That is, we’re using the distributive property twice.

Point out to students that when we multiply a binomial by a binomial, we’ll end up with four terms. It may be possible to create a trinomial by combining similar terms but there will be cases where there aren’t any similar terms, so we’ll just keep the four terms.

So at this point, the only thing left to check if we can combine like terms. You can also use a highlighter to highlight such terms.

(x + 8) (x – 2) = x²+ 2x + 8x + 16

So in this expression, we see that 2x and 8x are like terms and that we can simplify the expression by combining them, or in simple terms, adding them. By doing that, we’ll get the following trinomial:

x² + 10x + 16

Multiplying Binomials: Example 2

The above example was an example of two binomials with the same sign (addition) in between the terms of the binomials. But one of the most frequent things that confuse students tend to be cases with different signs in between the terms of the binomials. Provide such an example:

(5m – 1)(3m + 2) = ?

Draw the FOIL method on the whiteboard, again using different colors for each operation:

Explain to students that to avoid confusion, they can think of the binomials as two lists of two terms, each with a particular sign. Encourage the students not to think about operations in their answer, but just list each term with its sign. For instance (5m, -1) and (3m, 2).

It is useful to present the operations with each term in brackets from here.

In other words:

(5m – 1)(3m + 2) = (5m)(3m) + (5m)(2) + (-1)(3m) + (-1)(2) =

15m² + 10m + (-3m) + (-2) =

15m²+ 10m – 3m – 2 =

15m² + 7m – 2

Activities to Practice Multiplying Binomials

To implement this quiz activity in your classroom, you’ll need to prepare a PowerPoint with different binomial multiplication problems. Each slide contains one binomial problem that students have to solve in a given timeframe (for example, you can choose to give 5 minutes per problem).

You can use the binomial problems from this Exit Quiz Worksheet (Members Only) or from our free Assignment Worksheet . Make sure to also bring a timer (or alternatively, rely on your phone) to measure the time for a given problem.

Divide students into groups of 3, 4 and start playing! The group that has the most correct answers, in the end, wins the game. You can also introduce a symbolic prize for the quiz winners, such as a ‘no homework for a week’ prize!

If you’re a parent that’s homeschooling their child or children, you can easily adapt this activity by using the mentioned worksheets for an individual activity. Alternatively, if you have an older child as well, have them join in the quiz and add plenty of candy as prizes for the winner!

Before You Leave…

If you enjoyed the tips in this article, you’ll definitely love our lesson that goes in depth in teaching kids how to multiply binomials!

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This article is based on:

Unit 2 – Special Products and Factoring

• 2-1 Multiplying Binomials
• 2-2 Solving Two-Step Equations
• 2-3 Solving Multi-Step Equations
• 2-4 Solving Equations with Variables on Both Sides
• 2-5 Literal Equations and Formulas
• 2-6 Ratios, Rates, and Conversions
• 2-7 Solving Proportions
• 2-8 Proportions and Similar Figures
• 2-9 Percentages
• 2-10 Change Expressed as a Percent

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Multiplying Binomials Worksheets

There are several methods that you can use to multiply binomials, but by far the most valuable method (at the high school level) is the FOIL method. We will therefore focus our effort on centering our lessons around that method. The most important part of this is to make sure that you teach students the language of the FOIL method. What does each letter within the acronym stand for and how does that apply to binomials? It is also very important that students have mastered combining like terms before you start working with this skill. I would recommend a quick review of that skill before you jump in here. It will save both you and students a lot of headaches. This series of lesson and worksheets helps students learn how to find the product of two binomials.

Aligned Standard: HSA.APR.D.7

• Step-by-step Lesson - This lesson examines the basic all-around strategy for developing this.
• Solving Binomial Products Guided Lesson - We throw a bi- to tri- product problem in there too for some review.
• Guided Lesson Explanation - Once you understand how to do the first problem, it just gets easier from there.
• Multiplying Binomials Worksheet 1 - When you push these products together a foil-like method comes out.
• All Multiplication Skills Worksheet 2 - This sheet review the easy and the difficult problems all in one.
• Skill Review Packet - This is a pack of 4 worksheets that span the gamut of what kids really need to know.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

This set follows a stratified pattern of growth of ability on the materials for this section.

• Homework 1 - Find the end product of the two polynomials by multiplying them. This can normally be done in two steps, but a few may be a bit more extended.
• Homework 2 - Multiply the two binomials using proper technique. The first four problems will give you Deja Vu.
• Homework 3 - Another go around with this skill, but we extend the number of terms you have in the expressions.

Practice Worksheets

All the sheets in this series build in skill level and topic knowledge.

• Practice 1 - These might require two or three steps. You will get an idea the minute you start working with them.
• Practice 2 - Remember how to proceed here. Break it apart into several steps and make it work for yourself.
• Practice 3 - The values outside the parentheses is where to start. These can be a bit complicated.

Math Skill Quizzes

These should look pretty familiar to you by now.

• Quiz 1 - Pop these back over time. They will help you learn to break things down.
• Quiz 2 - I wish you could take these over time. Some a pretty simple, other not so much.
• Quiz 3 - Don't let the negative values sour you on these. Just remember a negative times a negative is a positive.

How to Multiply Binomials

The most common method of multiplying binomials is known as the FOIL method. It is not a necessary method used for multiplication in binomials by following the FOIL order, but using it definitely makes the process easier. The letters in the FOIL method refer to two terms (one of each of the two binomials) which are multiplied together in a certain specified order: First, outer, inner and last. If you following along with the diagram to the right: First (a x c), Outer (a x d), Inner (b x c), and Last (b x d).

For example: (2x + 3) (3x - 1). The following steps will demonstrate how the FOIL method can be used for binomial multiplicand.

1. Multiply the first term of every single binomial. (2x) (3x) = 6x 2

2. Now multiply the outer terms (2x) (-1) = -2x.

3. Multiply the inner terms with each other (3) (3x) = 9x

4. Multiply the last terms with each other (3) (-1) = -3

5. List down the results that have been achieved 6x 2 - 2x + 9x - 3.

6. Last step includes combining the terms that are similar. Right now, we have the following: 6x 2 | -2x | 9x | -3. We can see that -2x and 9x are like terms and can be combined to form 7x. The final value would be: 6x 2 + 7x - 3.

When Wil You Use This Skill in Real Life?

Binomials are often used in computer programming scenarios to model the use and execution of commands that the program is to complete. Architects often use the same math to calculate the heights of points in their projects. When economists are trying to determine the trends in the current state of the economy, they will often use binomials to describe events and probability values. During the election in process where often two candidates are seen as the only logical choices, we can use binomial distributions to examine the popularity and potential winner of the race. This math is often used to predict and track the movement of storm systems so that the weathermen can tell you when to wear shorts or bring an umbrella with you to school. As you can see the applications of this type of math is boundless.

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Multiplying Polynomials

A polynomial looks like this:

To multiply two polynomials:

• multiply each term in one polynomial by each term in the other polynomial
• add those answers together, and simplify if needed

Let us look at the simplest cases first.

1 term × 1 term   (monomial times monomial)

To multiply one term by another term, first multiply the constants , then multiply each variable together and combine the result, like this (press play):

(Note: I used "·" to mean multiply. In Algebra we don't like to use "×" because it looks too much like the letter "x")

For more about multiplying terms, read Multiply and Divide Variables with Exponents

1 term × 2 terms   (monomial times binomial)

Multiply the single term by each of the two terms, like this:

2 term × 1 terms   (binomial times monomial)

Multiply each of the two terms by the single term, like this:

(I did that one a bit faster by multiplying in my head before writing it down)

2 terms × 2 terms (binomial times binomial)

That is 4 different multiplications ... Why?

It is the same when we multiply binomials!

Instead of Alice and Betty, let's just use a and b , and Charles and David can be c and d :

We can multiply them in any order so long as each of the first two terms gets multiplied by each of the second two terms .

But there is a handy way to help us remember to multiply each term called " FOIL ".

It stands for " F irsts, O uters, I nners, L asts":

So you multiply the "Firsts" (the first terms of both polynomials), then the "Outers", etc.

Let us try this on a more complicated example:

2 terms × 3 terms (binomial times trinomial)

"FOIL" won't work here, because there are more terms now. But just remember:

Multiply each term in the first polynomial by each term in the second polynomial

And always remember to add Like Terms :

Example: (x + 2y)(3x − 4y + 5)

(x + 2y)(3x − 4y + 5)

= 3x 2 − 4xy + 5x + 6xy − 8y 2 + 10y

= 3x 2 + 2xy + 5x − 8y 2 + 10y

Note: −4xy and 6xy are added because they are Like Terms.

Also note: 6yx means the same thing as 6xy

Long Multiplication

You may also like to read about Polynomial Long Multiplication

Multiplying Three Binomials (A)

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Polynomial Worksheets

These worksheets focus on the topics typically covered in Algebra I

• Multiplying Monomials Worksheet
• Multiplying and Dividing Monomials Sheet
• Adding and Subtracting Polynomials worksheet
• Multiplying Monomials with Polynomials Worksheet
• Multiplying Binomials Worksheet
• Multiplying Polynomials
• Simplifying Polynomials
• Factoring Trinomials
• Operations with Polynomials Worksheet

Algebra 2 Polynomial Worksheets

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