4.7: Special Binomial Products
Three binomial products occur so frequently in algebra that we designate them as special binomial products . We have seen them before, but we will study them again because of their importance as time saving devices and in solving equations (which we will study in a later chapter).
These special products can be shown as the squares of a binomial
\((a+b)^2\) and \((a-b)^2\)
and as the sum and difference of two terms.
\[(a+b)(a-b) \nonumber\]
There are two simple rules that allow us to easily expand (multiply out) these binomials. They are well worth memorizing, as they will save a lot of time in the future.
Expanding \((a+b)^2\) and \((a−b)^2\)
To square a binomial:
- Square the first term.
- Take the product of the two terms and double it.
- Square the last term.
- Add the three results together
\((a+b)^2 = a^2 + 2ab + b^2\)
\((a-b)^2 = a^2 - 2ab + b^2\)
Expanding (a+b)(a−b)
To expand the sum and difference of two terms:†
- Square the first term and square the second term.
- Subtract the square of the second term from the square of the first term.
\((a+b)(a-b) = a^2 - b^2\)
Sample Set A
\((x+4)^{2}\)
Square the first term: \(x^{2}\).
The product of both terms is \(4x\). Double it: \(8x\).
Square the last term: 16.
Add them together: \(x^{2}+8x+16\)
\((x+4)^{2}=x^{2}+8 x+16\)
Note that \((x+4)^{2} \neq x^{2}+4^{2}\). The \(8x\) term is missing!
\((a-8)^{2}\)
Square the first term: \(a^{2}\).
The product of both terms is \(-8a\). Double it: \(-16a\).
Square the last term: 64.
Add them together: \(a^2 + (-16a) + 64\)
\[(a-8)^2 = a^2 - 16a + 64 \nonumber\]
Notice that the sign of the last term in this expression is “\(+\).” This will always happen since the last term results from a number being
squared
. Any nonzero number times itself is always positive.
\((+)(+) = +\) and \((-)(-) = +\)
The sign of the second term in the trinomial will always be the sign that occurs inside the parentheses.
\((y-1)^{2}\)
Square the first term: \(y^{2}\).
The product of both terms is \(-y\). Double it: \(-2y\).
Square the last term: +1.
Add them together: \(y^2 + (-2y) + 1\)
\((5x+3)^{2}\)
Square the first term: \(25x^{2}\).
The product of both terms is \(15x\). Double it: \(30x\).
Square the last term: 9.
Add them together: \(25x^2 + 30x + 9\)
\((7b-2)^{2}\)
Square the first term: \(49b^{2}\).
The product of both terms is \(-14b\). Double it: \(-28b\).
Square the last term: 4.
Add them together: \(49b^2 + (-28b) + 4\)
\((x+6)(x-6)\)
Square the first term: \(x^2\).
Subtract the square of the second term (\(36\)) from the square of the first term: \(x^2 - 36\)
\((x+6)(x-6) = x^2 - 36\)
\((4a−12)(4a+12)\)
Square the first term: \(16a^2\).
Subtract the square of the second term (\(144\)) from the square of the first term: \(16a^2-144\)
\((4a-12)(4a+12) = 16a^2 - 144\)
\((6x+8y)(6x−8y)\)
Square the first term: \(36x^2\).
Subtract the square of the second term (\(64y^2\)) from the square of the first term: \(36x^2 - 64y^2\)
\((6x+8y)(6x-8y) = 36x^2 - 64y^2\)
Practice Set A
Find the following products.
\((x+5)^2\)
- Answer
-
\(x^2 + 10x + 25\)
\((x+7)^2\)
- Answer
-
\(x^2 + 14x + 49\)
\((y-6)^2\)
- Answer
-
\(y^2 - 12y + 36\)
\((3a+b)^2\)
- Answer
-
\(9a^2 + 6ab + b^2\)
\((9m-n)^2\)
- Answer
-
\(81m^2 - 18mn + n^2\)
\((10x - 2y)^2\)
- Answer
-
\(100x^2 - 40xy + 4y^2\)
\((12a - 7b)^2\)
- Answer
-
\(144a^2 - 168ab + 49b^2\)
\((5h - 15k)^2\)
- Answer
-
\(25h^2 - 150hk + 225k^2\)
Exercises
For the following problems, find the products.
\((x+3)^2\)
- Answer
-
\(x^2 + 6x + 9\)
\((x+5)^2\)
\((x+8)^2\)
- Answer
-
\(x^2 + 16x + 64\)
\((x+6)^2\)
\((y+9)^2\)
- Answer
-
\(y^2 + 18y + 81\)
\((y+1)^2\)
\((a-4)^2\)
- Answer
-
\(a^2 - 8a + 16\)
\((a-6)^2\)
\((a-7)^2\)
- Answer
-
\(a^2 - 14a + 49\)
\((b+10)^2\)
\((b+15)^2\)
- Answer
-
\(b^2 + 30b + 225\)
\((a-10)^2\)
\((x-12)^2\)
- Answer
-
\(x^2 - 24x + 144\)
\((x+20)^2\)
\((y-20)^2\)
- Answer
-
\(y^2 - 40y + 400\)
\((3x + 5)^2\)
\((4x + 2)^2\)
- Answer
-
\(16x^2 + 16x + 4\)
\((6x - 2)^2\)
\((7x - 2)^2\)
- Answer
-
\(49x^2 - 28x + 4\)
\((5a - 6)^2\)
\((3a - 9)^2\)
- Answer
-
\(9a^2 - 54a + 81\)
\((3w - 2z)^2\)
\((5a - 3b)^2\)
- Answer
-
\(25a^2 - 30ab + 9b^2\)
\((6t - 7s)^2\)
\((2h - 8k)^2\)
- Answer
-
\(4h^2 - 32hk + 64k^2\)
\((a + \dfrac{1}{2})^2\)
\((a + \dfrac{1}{3})^2\)
- Answer
-
\(a^2 + \dfrac{2}{3}a + \dfrac{1}{9}\)
\((x + \dfrac{3}{4})^2\)
\((x + \dfrac{2}{5})^2\)
- Answer
-
\(x^2 + \dfrac{4}{5}x + \dfrac{4}{25}\)
\((x - \dfrac{2}{3})^2\)
\((y-\dfrac{5}{6})^2\)
- Answer
-
\(y^2 - \dfrac{5}{3}y + \dfrac{25}{36}\)
\((y + \dfrac{2}{3})^2\)
\((x + 1.3)^2\)
- Answer
-
\(x^2 + 2.6x + 1.69\)
\((x + 5.2)^2\)
\((a + 0.5)^2\)
- Answer
-
\(a^2 + a + 0.25\)
\((a + 0.08)^2\)
\((x - 3.1)^2\)
- Answer
-
\(x^2 - 6.2x + 9.61\)
\((y - 7.2)^2\)
\((b - 0.04)^2\)
- Answer
-
\(b^2 - 0.08b + 0.0016\)
\((f - 1.006)^2\)
\((x + 5)(x - 5)\)
- Answer
-
\(x^2 - 25\)
\((x+6)(x-6)\)
\((x+1)(x−1)\)
- Answer
-
\(x^2 - 1\)
\((t−1)(t+1)\)
\((f+9)(f−9)\)
- Answer
-
\(f^2 - 81\)
\((y−7)(y+7)\)
\((2y+3)(2y−3)\)
- Answer
-
\(4y^2 - 9\)
\((5x+6)(5x−6)\)
\((2a−7b)(2a+7b)\)
- Answer
-
\(4a^2 - 49b^2\)
\((7x+3t)(7x−3t)\)
\((5h−2k)(5h+2k)\)
- Answer
-
\(25h^2 - 4k^2\)
\((x + \dfrac{1}{3})(x - \dfrac{1}{3})\)
\((a + \dfrac{2}{9})(a - \dfrac{2}{9})\)
- Answer
-
\(a^2 - \dfrac{4}{81}\)
\((x + \dfrac{7}{3})(x - \dfrac{7}{3})\)
\((2b + \dfrac{6}{7})(2b - \dfrac{6}{7})\)
- Answer
-
\(4b^2 - \dfrac{36}{49}\)
Expand \((a+b)^2\) to prove it is equal to \(a^2 + 2ab + b^2\).
Expand \((a-b)^2\) to prove it is equal to \(a^2 - 2ab + b^2\).
- Answer
-
\((a-b)(a-b) = a^2 - ab - ab + b^2 = a^2 - 2ab + b^2\)
Expand \((a+b)(a-b)\) to prove it is equal to \(a^2-b^2\).
Fill in the missing label in the equation below.
- Answer
-
First term squared
Label the parts of the equation below.
Label the parts of the equation below.
- Answer
-
a) Square the first term.
b) Square the second term and subtract it from the first term.
Exercises for Review
Simplify \((x^3y^0z^4)^5\).
Find the value of \(10^{-1} \cdot 2^{-3}\)
- Answer
-
\(\dfrac{1}{80}\)
Find the product.
\((x+6)(x-7)\).
Find the product.
\((5m - 3)(2m + 3)\)
- Answer
-
\(10m^2 + 9m - 9\)
Find the product.
\((a+4)(a^2 - 2a + 3)\)