What's the point of memorizing these patterns? I think it's better just to solve the problems instead of memorizing some sort of pattern.

For multiplication, you're right... it can be just as easy to just multiply the binomials like any other binomials. However, learning the patterns will help you later when you learn how to: 1) Factor polynomials 2) Solve quadratic equations by completing the square 3) Completing the square to work with equations for circles

How would I cube a polynomial?

square it first, then multiply the square times the polynomial, it could get complicated, but doable

I'm studying for my teacher certification, and just went through as many videos on polynomials as I could. On my practice test, there was a problem I hope to get explained. It read "(3+2i)(4+3i)" and the answer was "6+17i". How did the 6i^2 end up cancelled out and subtracting 6 as well? Very confused. Does it matter if the letters are after the constants in the problem?

Remember, i = sqrt(-1) i^2 = sqrt(-1) * sqrt(-1) = -1 Thus, 6i^2 = 6(-1) = -6 Hope this helps.

What would you do if you were trying to do (8x-5)^5 and your calculator says: Overflow Error

x can be any number, so the calculator doesn't know which number equals x, so x can be any number, so then the equation then can equal anything. If you wanted to put it in standard form, you would: see below.↓.Hope this helps.

what is the best way to remember how to add and subtract or multiple and divide negatives

Like everything else in mathematics, the key is practice. Instead of just memorizing the formula, if you solve enough of these, the answers start to jump out. The advantage is that you'll then have a pattern of understanding that will make future lessons easier.

why do we need this if we forget it when we are in our 40s

I am in my 60s and I still remember it, if you do not put any value on what you are learning, you may forget it in your 20s.

I cant see the patterns! The only way I can solve these is by using the distributive property. Anyone else have this issue?

It will come with time as you continue to solve them by using the distributive property. When it does, you will just be able to solve them a little more quickly.

Main content

College Algebra

Course: College Algebra > Unit 4

Lesson 2: Special products of binomials

Binomial special products review

Google Classroom

A review of the difference of squares pattern (a+b)(a-b)=a^2-b^2, as well as other common patterns encountered while multiplying binomials, such as (a+b)^2=a^2+2ab+b^2.

These types of binomial multiplication problems come up time and time again, so it's good to be familiar with some basic patterns.

The "difference of squares" pattern:

(a + b) (a - b) = a^{2} - b^{2}

Two other patterns:

\begin{aligned} (a + b)^{2} = a^{2} + 2 a b + b^{2} \\ (a - b)^{2} = a^{2} - 2 a b + b^{2} \end{aligned}

Example 1

Expand the expression.

(c - 5) (c + 5)

The expression fits the difference of squares pattern:

(a + b) (a - b) = a^{2} - b^{2}

So our answer is:

(c - 5) (c + 5) = c^{2} - 25

But if you don't recognize the pattern, that's okay too. Just multiply the binomials as normal. Over time, you'll learn to see the pattern.

\begin{aligned} (c - 5) (c + 5) \\ = & c (c) + c (5) - 5 (c) - 5 (5) \\ = & c (c) + 5 c - 5 c - 5 (5) \\ = & c^{2} - 25 \end{aligned}

Notice how the "middle terms" cancel.

Want another example? Check out this video.

Example 2

Expand the expression.

(m + 7)^{2}

The expression fits this pattern:

(a + b)^{2} = a^{2} + 2 a b + b^{2}

So our answer is:

(m + 7)^{2} = m^{2} + 14 m + 49

But if you don't recognize the pattern, that's okay too. Just multiply the binomials as normal. Over time, you'll learn to see the pattern.

\begin{aligned} (m + 7)^{2} \\ = & (m + 7) (m + 7) \\ = & m (m) + m (7) + 7 (m) + 7 (7) \\ = & m (m) + 7 m + 7 m + 7 (7) \\ = & m^{2} + 14 m + 49 \end{aligned}

Want another example? Check out this video.

Example 3

Expand this expression.

(6 w - y) (6 w + y)

The expression fits the difference of squares pattern:

(a + b) (a - b) = a^{2} - b^{2}

So our answer is:

\begin{aligned} (6 w - y) (6 w + y) \\ = & (6 w)^{2} - y^{2} \\ = & 36 w^{2} - y^{2} \end{aligned}

But if you don't recognize the pattern, that's okay too. Just multiply the binomials as normal. Over time, you'll learn to see the pattern.

\begin{aligned} (6 w - y) (6 w + y) \\ = & 6 w (6 w) + 6 w (y) - y (6 w) - y (y) \\ = & 6 w (6 w) + 6 w y - 6 w y - y (y) \\ = & 36 w^{2} - y^{2} \end{aligned}

Notice how the "middle terms" cancel.

Want more practice? Check out this intro exercise and this slightly harder exercise.

Want to join the conversation?

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Omor Almamun
Posted 6 years ago. Direct link to Omor Almamun's post “What's the point of memor...”
What's the point of memorizing these patterns? I think it's better just to solve the problems instead of memorizing some sort of pattern.
Button navigates to signup pageComment on Omor Almamun's post “What's the point of memor...”
(30 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “For multiplication, you'r...”
  For multiplication, you're right... it can be just as easy to just multiply the binomials like any other binomials. However, learning the patterns will help you later when you learn how to:
  1) Factor polynomials
  2) Solve quadratic equations by completing the square
  3) Completing the square to work with equations for circles
  Button navigates to signup page
  (64 votes)
Bianca
Posted 6 years ago. Direct link to Bianca's post “How would I cube a polyno...”
How would I cube a polynomial?
Button navigates to signup pageComment on Bianca's post “How would I cube a polyno...”
(15 votes)
Answer
- David Severin
  Posted 6 years ago. Direct link to David Severin's post “square it first, then mul...”
  square it first, then multiply the square times the polynomial, it could get complicated, but doable
  Button navigates to signup page
  (15 votes)
cnitzel
Posted 6 years ago. Direct link to cnitzel's post “I'm studying for my teach...”
I'm studying for my teacher certification, and just went through as many videos on polynomials as I could. On my practice test, there was a problem I hope to get explained. It read "(3+2i)(4+3i)" and the answer was "6+17i". How did the 6i^2 end up cancelled out and subtracting 6 as well? Very confused. Does it matter if the letters are after the constants in the problem?
Button navigates to signup pageComment on cnitzel's post “I'm studying for my teach...”
(9 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “Remember, i = sqrt(-1) i^...”
  Remember, i = sqrt(-1)
  i^2 = sqrt(-1) * sqrt(-1) = -1
  Thus, 6i^2 = 6(-1) = -6
  Hope this helps.
  Button navigates to signup page
  (17 votes)
phillip wollmann
Posted 3 years ago. Direct link to phillip wollmann's post “why do we need this if we...”
why do we need this if we forget it when we are in our 40s
Button navigates to signup pageComment on phillip wollmann's post “why do we need this if we...”
(4 votes)
Answer
- David Severin
  Posted 3 years ago. Direct link to David Severin's post “I am in my 60s and I stil...”
  I am in my 60s and I still remember it, if you do not put any value on what you are learning, you may forget it in your 20s.
  Comment on David Severin's post “I am in my 60s and I stil...”
  (23 votes)
rh805331
Posted 4 years ago. Direct link to rh805331's post “if you solve enough of th...”
if you solve enough of these, the answers start to jump out. The advantage is that you'll then have a pattern of understanding that will make future lessons easier.
Button navigates to signup pageButton navigates to signup page
(11 votes)
Answer
Dado
Posted 4 years ago. Direct link to Dado's post “What would you do if you ...”
What would you do if you were trying to do (8x-5)^5 and your calculator says: Overflow Error
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- timotime12
  Posted 4 years ago. Direct link to timotime12's post “x can be any number, so t...”
  x can be any number, so the calculator doesn't know which number equals x, so x can be any number, so then the equation then can equal anything.
  If you wanted to put it in standard form, you would:
  see below.↓.Hope this helps.
  Button navigates to signup page
  (6 votes)
lharty1323
Posted 4 years ago. Direct link to lharty1323's post “what is the best way to r...”
what is the best way to remember how to add and subtract or multiple and divide negatives
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Bill Zemon
  Posted 4 years ago. Direct link to Bill Zemon's post “Like everything else in m...”
  Like everything else in mathematics, the key is practice. Instead of just memorizing the formula, if you solve enough of these, the answers start to jump out. The advantage is that you'll then have a pattern of understanding that will make future lessons easier.
  Button navigates to signup page
  (3 votes)
Addie Ledbetter
Posted 9 months ago. Direct link to Addie Ledbetter's post “I cant see the patterns! ...”
I cant see the patterns! The only way I can solve these is by using the distributive property. Anyone else have this issue?
Button navigates to signup pageComment on Addie Ledbetter's post “I cant see the patterns! ...”
(3 votes)
Answer
- Timothy S. Moran
  Posted 7 months ago. Direct link to Timothy S. Moran's post “It will come with time as...”
  It will come with time as you continue to solve them by using the distributive property. When it does, you will just be able to solve them a little more quickly.
  Button navigates to signup page
  (6 votes)
isalia
Posted 4 years ago. Direct link to isalia's post “Can't you just use the ar...”
Can't you just use the are model and get the same answer?
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(4 votes)
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jonathan.postelnik
Posted 7 years ago. Direct link to jonathan.postelnik's post “Why is (a+b)^2 quadratics...”
Why is (a+b)^2 quadratics form if this is polynomials?
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(2 votes)
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