Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.
Using the Distributive Property
We will start by using the Distributive Property. Look again at Example \(\PageIndex{6}\).
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We distributed the p to get |
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What if we have (x + 7) instead of p? Think of the (x + 7) as the \(\textcolor{red}{p}\) above. |
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Distribute (x + 7). |
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Distribute again. |
x2 + 7x + 3x + 21 |
Combine like terms. |
x2 + 10x + 21 |
Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.
Be careful to distinguish between a sum and a product.
\[\begin{split} &\textbf{Sum} \qquad \qquad \qquad \quad \textbf{Product} \\ &x + x \qquad \qquad \qquad \qquad x \cdot x \\ &\; \; 2x \qquad \qquad \qquad \qquad \qquad x^{2} \\ combine\; &like\; terms \qquad add\; exponents\; of\; like\; bases \end{split}\]
Example \(\PageIndex{7}\):
Multiply: (x + 6)(x + 8).
Solution
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Distribute (x + 8). |
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Distribute again. |
x2 + 8x + 6x + 48 |
Simplify. |
x2 + 14x + 48 |
Exercise \(\PageIndex{13}\):
Multiply: (x + 8)(x + 9).
- Answer
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\(x^{2}+17 x+72 \)
Exercise \(\PageIndex{14}\):
Multiply: (a + 4)(a + 5).
- Answer
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\(a^{2}+9 a+20 \)
Now we'll see how to multiply binomials where the variable has a coefficient.
Example \(\PageIndex{8}\):
Multiply: (2x + 9)(3x + 4).
Solution
Distribute (3x + 4). |
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Distribute again. |
6x2 + 8x + 27x + 36 |
Simplify. |
6x2 + 35x + 36 |
Exercise \(\PageIndex{15}\):
Multiply: (5x + 9)(4x + 3).
- Answer
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\(20 x^{2}+51 x+27 \)
Exercise \(\PageIndex{16}\):
Multiply: (10m + 9)(8m + 7).
- Answer
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\(80 m^{2}+142 m+63 \)
In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.
Example \(\PageIndex{9}\):
Multiply: (4y + 3)(6y − 5).
Solution
Distribute. |
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Distribute again. |
24y2 − 20y + 18y − 15 |
Simplify. |
24y2 − 2y − 15 |
Exercise \(\PageIndex{17}\):
Multiply: (7y + 1)(8y − 3).
- Answer
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\( 56 y^{2}-13 y-3\)
Exercise \(\PageIndex{18}\):
Multiply: (3x + 2)(5x − 8).
- Answer
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\(15 x^{2}-14 x-16 \)
Up to this point, the product of two binomials has been a trinomial. This is not always the case.
Example \(\PageIndex{10}\):
Multiply: (x + 2)(x − y).
Solution
Distribute. |
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Distribute again. |
x2 - xy + 2x - 2y |
Simplify. |
There are no like terms to combine. |
Exercise \(\PageIndex{19}\):
Multiply: (x + 5)(x − y).
- Answer
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\( x^{2}-x y+5 x-5 y\)
Exercise \(\PageIndex{20}\):
Multiply: (x + 2y)(x − 1).
- Answer
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\( x^{2}-x+2 x y-2 y\)