Multiplying Binomials and Squaring Binomials


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Example 5

Use an area model to multiply \(22 \times 86\). 

\(\)

 

Example 5 Continued 

Use an area model to multiply \(22 \times 86\). 

Solution 

We divide the rectangle into smaller portions. 

 

Example 5 Continued

Use an area model to multiply \(22 \times 86\).

Alternative Solution 

The way we split the larger rectangle is not the only way to use an area model.

 

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Explore This 1


Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/kxjvajt2#


Explore This Summary

In this activity, you may have noticed the following:

  • The dimensions of the rectangle were always \(x+r\), and \(x+s\).
    • \(r\) and \(s\) were the values on the sliders.
  • Since the area of a rectangle is calculated by multiplying the length by the width, one way of expressing the total area was by multiplying \(x+r\) and \(x+s\), i.e., \((x+r)(x+s)\).
    • This was the factored version of the area.
  • The area could also be calculated by multiplying the lengths and widths of the four smaller rectangles and adding them together.
    • This was the expanded version of the area.

A rectangle with length x plus 4 and width x plus 2. The total area can be found by summing the areas of the divided rectangle.


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Multiplying Two Binomials

Consider the expression \((a+b)(c+d)\). 

 

 

Example 6 — Part A

Expand and simplify the following:

  1. ​​​​\((x+3)(x+6)\) 
  2. \((3x-1)(2x-5)\)

 

 

Example 6 — Part B

Expand and simplify the following:

  1. ​​​​\((x+3)(x+6)\) 
  2. \((3x-1)(2x-5)\). 

 

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Example 7

Expand and simplify \((7x+4)(7x-4)\). 

Solution  

A table can help organize which pairs of terms need to be multiplied.  

  \(7x\) \(-4\)
\(7x\) \(7x(7x)=49x^2\) \(7x(-4)=-28x\)
\(4\) \(4(7x)=28x\) \(4(-4)=-16\)

You can also go straight to the algebraic solution:

\(\begin{align*} (\class{hl2}{7x} + \class{hl2}{4})(\class{hl4}{7x}- \class{hl4}{4}) & = \class{hl2}{7x}(\class{hl4}{7x}) + \class{hl2}{7x}(\class{hl4}{-4}) + \class{hl2}{4}(\class{hl4}{7x}) +\class{hl2}{4}(\class{hl4}{-4}) \\ &= 49x^2 -28x+28x -16 \\ &= 49x^2 -16 \end{align*}\) 

Notice that the \(x\)-terms added to \(0x\), or cancelled each other out. 


Check Your Understanding 2


Expand and simplify the following expression.

\(((((((exp(r))*(e))*(s))*(s))*(i))*(o))*(n)\)

Enter \(a^b\) as "\(a^{\wedge} b\)".

There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:

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Example 8

Expand and simplify \((x+9)^2\).

 

 

Squaring Any Binomial

 

 

Example 9

Expand and simplify \((3x-2)^2\).

 

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Check Your Understanding 3


Expand and simplify the following expression.

\(((((((exp(r))*(e))*(s))*(s))*(i))*(o))*(n)^{2}\)

Enter \(a^b\) as "\(a^{\wedge} b\)".

There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:

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Try This Revisited

A square has side lengths of \(w\) units. The square's side lengths are then increased by \(3\) units on each side. 

Square 1

A square with width w.

Square 2

A square with width w plus 3.

Let \(\Delta A\) represent the difference in the areas of the two squares (Square 2 minus Square 1). 

 

For example, when \(w=2\), the area of Square 1 is \(2^2=4\).

Then, the side lengths of Square 2 are \(5\) units, and the area of this square is \(5^2=25\). 

So, for \(w=2\), \(\Delta A = 25-4\) , or \(\Delta A = 21\). 

Determine a simplified equation expressing \(\Delta A\) in terms of \(w\). 

That is, find an equation that allows you to calculate the difference in areas directly from the original side length without having to find the two areas first. 

Solution

The area of Square 1 is \(w \times w\), or \(w^2\). 

The area of Square 2 is \((w+3)^2\).

Thus, we can express the difference in the areas as \(\Delta A = (w+3)^2 - w^2\).

This equation can be simplified by following order of operations. We will expand \((w+3)^2\) first, then subtract \(w^2\).

\(\begin{align*} \Delta A &= (w+3)^2 - w^2 \\ \Delta A &= w^2 + 6w + 9 - w^2 \\ \Delta A &= 6w+9 \end{align*}\)

Note that \(w^2-w^2\) simplified to \(0w^2\).

Our simplified equation relating the difference in area to the width is \(\Delta A = 6w+9 \). 

We can substitute any width into this equation, and it will tell us how much the area of the square will increase if this width is increased by \(3\).

Check: 

Earlier, we calculated \(\Delta A\) directly for \(w=2\), and arrived at an answer of \(\Delta A = 21\). Let's check if our new equation also gives this result.

Substitute \(w=2\) into \(\Delta A = 6w+9\): 

\(\begin{align*} \Delta A &= 6(2)+9 \\ &= 12 + 9 \\ &= 21 \end{align*}\)

The result from the equation is consistent with our earlier calculation. 

Example 10

Expand the following:

  1. \((2x-y)(x+3y)\)
  2. \((4x-5y)^2\)

Solution — Part A

Use the distributive property to multiply the two binomials.

Recall that \(x \times y = xy\). So, \((2x)(3y)=6xy\). 

\(\begin{align*}(2x-y)(x+3y) &= 2x^2+6xy-xy-3y^2 \\ &= 2x^2+5xy-3y^2\end{align*}\)   

Optional Table

  \(x\) \(3y\)
\(2x\) \(2x^2\) \(6xy\)
\(-y\) \(-xy\) \(-3y^2\)

Solution — Part B

Use \((a+b)^2 = a^2+2ab+b^2 \) with \(a=4x\) and \(b=-5y\). 

\(\begin{align*}(4x-5y)^2 &= (4x)^2+2(4x)(-5y) + (-5y)^2 \\ &= 16x^2-40xy+25y^2\end{align*}\)

Alternate solution: Write \((4x-5y)^2 = (4x-5y)(4x-5y)\) and expand using the distributive property.


Check Your Understanding 4


Expand and simplify the following expressions.

Enter \(a^b\) as "\(a^{\wedge} b\)".

  1. \((3x - 5y)(2x - y) =\) There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:
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  2. \((x + 7y)^{2} = \) There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:
    null

  3. \((6x - 4y)(6x + 4y) =\) There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:
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