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Binomial special products review

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:
left parenthesis, a, plus, b, right parenthesis, left parenthesis, a, minus, b, right parenthesis, equals, a, squared, minus, b, squared
Two other patterns:
\begin{aligned} &(a+b)^2=a^2+2ab+b^2\\\\ &(a-b)^2=a^2-2ab+b^2 \end{aligned}

Example 1

Expand the expression.
left parenthesis, c, minus, 5, right parenthesis, left parenthesis, c, plus, 5, right parenthesis
The expression fits the difference of squares pattern:
left parenthesis, a, plus, b, right parenthesis, left parenthesis, a, minus, b, right parenthesis, equals, a, squared, minus, b, squared
left parenthesis, c, minus, 5, right parenthesis, left parenthesis, c, plus, 5, right parenthesis, equals, c, squared, minus, 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} &(\purpleD{c-5})(c+5)\\\\ =&\purpleD{c}(c)+\purpleD{c}(5)\purpleD{-5}(c)\purpleD{-5}(5)\\\\ =&\purpleD{c}(c)+\redD{5c-5c}\purpleD{-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.
left parenthesis, m, plus, 7, right parenthesis, squared
The expression fits this pattern:
left parenthesis, a, plus, b, right parenthesis, squared, equals, a, squared, plus, 2, a, b, plus, b, squared
left parenthesis, m, plus, 7, right parenthesis, squared, equals, m, squared, plus, 14, m, plus, 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\\\\ =&(\blueD{m+7})(m+7)\\\\ =&\blueD{m}(m)+\blueD{m}(7)+\blueD{7}(m)+\blueD{7}(7)\\\\ =&\blueD{m}(m)\greenD{+7m+7m}+\blueD{7}(7)\\\\ =&m^2+14m+49 \end{aligned}
Want another example? Check out this video.

Example 3

Expand this expression.
left parenthesis, 6, w, minus, y, right parenthesis, left parenthesis, 6, w, plus, y, right parenthesis
The expression fits the difference of squares pattern:
left parenthesis, a, plus, b, right parenthesis, left parenthesis, a, minus, b, right parenthesis, equals, a, squared, minus, b, squared
\begin{aligned} &(6w-y)(6w+y) \\\\ =&(6w)^2-y^2 \\\\ =&36w^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} &(\purpleD{6w-y})(6w+y)\\\\ =&\purpleD{6w}(6w)+\purpleD{6w}(y)\purpleD{-y}(6w)\purpleD{-y}(y)\\\\ =&\purpleD{6w}(6w)+\redD{6wy-6wy}\purpleD{-y}(y)\\\\ =&36w^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?

• 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.
• If you consider why these patterns are true (for example by going through the process of multiplying out the binomials in each pattern), then you will have an easier time remembering them.

The following videos of geometric area models can also help you remember these patterns.

• How would I cube a polynomial?
• square it first, then multiply the square times the polynomial, it could get complicated, but doable
• 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.
• 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.