What is the product of (3n+2)(n+3)?

Question

This question asks you to multiply binomials. (A binomial is the sum of two terms.)

Answer 1

To multiply binomials (multiplying two terms that each have two terms), we typically talk about using the FOIL method:

where you multiply the first and first terms, the two outside terms, the two inside terms, and the two last terms and then add all 4 together.

But you can also use the rectangle method which can EASILY be expanded to include many terms instead of just two:

This involves putting each binomial on the outside of a rectangle, one on top and one along the side. Divide to rectangle into sections to represent the number of terms along the top or side (in this case, two on top, two on the side). Then multiply each part, putting each answer in the corresponding "mailbox slot". Then add them all together.

Answer 2

Debbie's answer is fantastic, so I would look to hers on how to calculate (3n+2)(n+3). But I want to talk about why the FOIL method is the valid answer. I'll do it in three ways: an intuitive explanation, a geometric explanation, and a technical explanation:

First, some common background: One of the most important ideas in math is the distribute property. In algebra, this idea appears as: m(a+b) = ma + ba. The distribute property is vital to why FOIL-ing works.

Intuitive Explanation:

In the math statement (3n+2)(n+3), one thing to remember is that we're really just mutliplying two numbers together. It looks like a complex statement, but (3n+2) is a number, for 3 times a number n is a number, and then adding 2 still creates a number. Similarly, n+3 is a number. Since 3n+2 is a number, we can give it a name; let m = 3n+2. Then the math statement (the technical term for "math statement" is an expression) becomes:

(3n+2)(n+3) = m(n+3) = mn + m3

by the distribute property! But hold on: m = (3n+2). So we can substitute back and have

mn + m3 = (3n+2)n + (3n+2)3 = 3n² + 2n + 9n + 6 = 3n² + 11n + 9,

the exact answer Debbie found.

So while FOIL (First two terms, Outer terms, Inner terms, Last two terms) is a helpful memory trick for doing the multiplication fast, the reason FOIL works is because even complicated-looking addition like (3n+2) is actually just a number, so the following math is valid:

(3n+2)(n+3) = (3n+2)n + (3n+2)3 = 3n² + 2n + 9n + 6 = 3n² + 11n + 9

and doing it the other way,

(3n+2)(n+3) = 3n(n+3) + 2(n+3) = 3n² + 9n + 2n + 6 = 3n² + 11n + 9.

Geometric Explanation:

(3n+2)(n+3) can be thought of as the following rectangle:

. . . . .3n . . . 2

. . .--------------------

. n | 3n² . | . 2n |

. . .|______|_____ |

. 3 | 9n . .| . 6 . .|

. . .``````````````````````

If you add the areas of each of these sub- rectangles together, you'll get 3n² + 11n + 9. Pretty cool to see the distributive property in action visually, right?

Technical Explanation:

This explanation is for the very math-savvy, so feel free to duck out now if this explanation goes over your head!

When we look at the expression (3n+2)(n+3), we're dealing with the multiplication of numbers, specifically real numbers. Intuitively, real numbers are every decimal number you can think of, but the exact technical definition of the real numbers is complicated and above the scope of this explanation, ha ha.

The real numbers are closed under multiplication and addition, which means that multiplication of real numbers always produces a real number, and addition of real numbers always produces a real number.. This mean 3n is a real number (3 * n), so 3n + 2 is a real number as we're adding a real number. The real numbers also have the distribute property, described earlier, which is that m(a+b) = ma + mb. So since (3n+2) is a real number as shown earlier, and since n and 3 are both real numbers, by the distributive property we have:

(3n+2)(n+3) = (3n+2) * n + (3n+2) * 3,

which leads to the same result as shown earlier.

So FOIL works because it is the only possible result of (3n+2)(n+3) that complies with the axioms of the real numbers. It's the only logical deduction.

Why are the axioms of the real numbers true? That's a deeper question, and gets to deep math nerdy territory. The axioms of the real numbers, like the distributive property, match our intuitions on how objects should combine together. Mathematicians created this concept of real numbers because they matched our intutions, and they gave this object, called a set, said properties we know and love because they are the properties we wanted. So the answer to why FOIL works is that it's a logical deduction to axioms mathematicians decided were useful to be true, yet those axioms match the natural way we combine objects together.

Answer 3

The easiest way to solve this problem is to simply use the acronym FOIL where F stands for first, O stands for outer, I stands for inside, and L stands for last. The way you apply this acronym to the factored form of a parabula is seen in this simple example

Let us say we have the simple equation

(2+x)(3+y)

The first (Foil) terms in the two parts that are in parentheses are 2 and 3, so our first step would be to multiply 2 by 3. Now the outer (fOil) terms of the entire equation, or the the terms that are farthest from eachother, are 2 and y, so we would multiply 2 by y. The inner (foIl) terms of the entire equation, or the terms that are closes to eachother without a + between them are x and 3, so we would multiply x by 3. Finally, the last (foiL) terms, or the ones that are the second terms in each parenthesized section, are x and y. So, we would multiply x by y. After all of this we would get

6 +2y +3x +xy

Now that we have a basic understanding of FOIL, lets apply it to the problem we have been given

(3n +2)(n +3)

(3n)(n) + (3n)(3) + (2)(n) + (2)(3)

F O I L

3n^2 + 9n +2n +6

3n^2 + 11n +6 is are answer

Answer 4

When you multiply binomials you must multiply (distribute) each term from the first set of parenthesis to each term in the second set of parenthesis. The best way to stay organized is to use the FOIL method, which stands for First, Outside, Inside, Last.

(3n+2)(n+3)

First of each parenthesis (3n)(n) = 3n²

Outside (1st and last) (3n)(3)= 9n

Inside (middle terms) (2)(n) = 2n

Last of each parenthesis (2)(3) = 6

(3n+2)(n+3) = 3n² + 9n + 2n + 6

First+ Outside+Inside+Last

Since the 2 middle terms (9n and 2n) can be combined (like terms have the same variable to the same power), the answer is a trinomial:

3n² + 9n +2n + 6 =

3n² + 11n +6

Challenge: You do not always get a trinomial. Can you create an example where you would not end up with a trinomial as the product?

Answer 5

Answer 6

Hello,

I have worked out this problem in this video. When you can, check it out and let me know if you have any additional questions. I hpoe you found this to be helpful. Take care!

Dr. Christal-Joy Turner