Tutorial: Multiplying algebraic expressions

Tutorial: Multiplying and factoring algebraic expressions

This tutorial: Part A: Multiplying algebraic expressions

Go to Part B: Factoring algebraic expressions: Taking out common factors

%%Note If you are happy with multiplying algebraic expressions and want to study factoring, go to Part B by clicking on its link above.

Single step calculations

One of the most important mathematical tools for multiplying algebraic expressions is the distributive law for real numbers:

#[Distributive law for real numbers][Ley distributiva para los números reales]#

#[If $a,$ $b,$ and $c$ are any real numbers, then:][Si $a, b,$ y $c$ son cualquieras números reales, entonces:]#

#[Law][Ley]#	Examples
$a(b \pm c) = ab \pm ac$ #[Left distributive law][Ley distributiva izquierda]#	$x(x+1)$\t = $x(x) + x(1) $ \\ \t ${}=x^2 + x$ $x^3(y-x)$\t ${}= x^3(y) - x^3(x) $ \\ \t ${}=x^3y - x^4$ $-7(x+y+z)$ ${}= (-7)x + (-7)y + (-7)z$ * ${}=-7x - 7y - 7z$ $\dfrac{4x^2}{3y}\left(\dfrac{xy^2}{2z} - \dfrac{y}{x}\right)$ ${}= \dfrac{4x^2}{3y}\left(\dfrac{xy^2}{2z}\right) - \dfrac{4x^2}{3y}\left(\dfrac{y}{x}\right)$ ${}= \dfrac{4x^2\cdot xy^2}{3y\cdot 2z} - \dfrac{4x^2\cdot y}{3y\cdot x}$ #[Rule for multiplying fractions][Relga para multiplicar fracciones]# ${}= \dfrac{4x^3y^2}{6yz} - \dfrac{4x^2y}{3xy}$ #[Rules for exponents][Relgas para exponentes]# ${}= \dfrac{2x^3y}{3z} - \dfrac{4x}{3}$
$(a \pm b)c = ac \pm bc$ #[Right distributive law][Ley distributiva derecha]#	$(x+1)y$\t ${}=x(y) + 1(y) $ \\ \t ${}=xy + y$ $(1-3x+x^2)x$ ${}=1(x) - 3x(x) + x^2(x)$ ${}=x - 3x^2 + x^3$ $\left(\dfrac{xy^2}{2z} + \dfrac{2y}{3x}\right)\dfrac{z}{xy^2}$ ${}= \dfrac{xy^2}{2z}\left(\dfrac{z}{xy^2}\right) + \dfrac{2y}{3x}\left(\dfrac{z}{xy^2}\right)$ ${}= \dfrac{xy^2\cdot z}{2z\cdot xy^2} + \dfrac{2y\cdot z}{3x\cdot xy^2}$ #[Rule for multiplying fractions][Relga para multiplicar fracciones]# ${}= \dfrac{xy^2z}{2xy^2z} + \dfrac{2yz}{3x^2y^2}$ #[Rules for exponents][Relgas para exponentes]# ${}= \dfrac{1}{2} + \dfrac{2z}{3x^2y}$

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* #[We can justify the use of the distributive law with three (or more) summands in the parentheses by arguing as follows:][ Podemos justificar el uso de la ley distributiva con tres (o más) sumandos en los paréntesis como sigue:]#

$a(b+c+d)$ \t ${}= a(b+[c+d])\quad$ \t #[Think of $b+c+d$ as ][Piensa en $b+c+d$ como]# $b+[c+d].$ \\ \t ${}= ab+a[c+d]$ \t #[Distributive law for two summands][Ley distributiva para tres sumandos]# \\ \t ${}= ab+ac+ac$ \t #[Distributive law for two summands again][Ley distributiva para tres sumandos otra vez]#

Multi-step calculations

If what we are distributing is itself a sum or difference, then it is necessary to apply the distributive law more than once:

Examples

$\color{crimson}{(2x+1)}(3x-2)$ \t ${}=\color{crimson}{(2x+1)}(3x)\ \ + \ \ \color{crimson}{(2x+1)}(-2)$ \gap[40] \t #[Distribute the ][Distribuya la]# $\color{crimson}{(2x+1)}$ \\ \t ${}=6x^2 + 3x \ \ - \ 4x - 2$ \t #[Apply the distributive law to each summand.][Aplica la ley distributiva a cada sumando.]# \\ \t ${}=6x^2 - x - 2$ \t #[Simplify (combine the $3x$ and $-4x$).][Simplifica (combina la $3x$ y la $-4x$).]# \\ \\ $\color{crimson}{(1 - y)}(1 + y - y^2)$ \t ${}=\color{crimson}{(1 - y)}(1)\ \ + \ \ \color{crimson}{(1 - y)}(y)\ \ - \ \ \color{crimson}{(1 - y)}(y^2)$ \t #[Distribute the ][Distribuya la]# $\color{crimson}{(1 - y)}$ \\ \t ${}=1 - y\ \ + \ \ y - y^2\ \ - \ \ (y^2 - y)$ \t #[Apply the distributive law to each summand.][Aplica la ley distributiva a cada sumando.]# \\ \t ${}=1 - y \ \ + \ \ y - y^2\ \ - \ \ y^2 + y$ \t #[Distribute the minus sign.][Distribuya el signo menos.]# \\ \t ${}=1 + y - 2y^2$ \t #[Simplify.][Simplifica.]#

FOIL Method

There is quicker way of expanding expressions such as the first and second one above, called the "FOIL" method.

%%FOIL

The FOIL method is used to expand products of the form

$(a + b)(c + d)$

#[as follows:][como sigue:]#

\\ \t #[F][P]# \t #[First][Primeros]# \t #[Multiply the first terms.][Multiplica los términos primeros]# \t $\color{#aaaaaa}{(\bold{\color{#c1026f}{a}} + b)(\bold{\color{#c1026f}{c}} + d)}$ \t $\color{#c1026f}{a \times c = ac}$ \\ \t #[O][EX]# \t #[Outer][EXternos]# \t #[Multiply the outer terms.][Multiplica los términos externos]# \t $\color{#aaaaaa}{(\bold{\color{#0ea05e}{a}} + b)(c + \bold{\color{#0ea05e}{d}})}$ \t $\color{#0ea05e}{a \times d = ad}$ \\ \t #[I][IN]# \t #[Inner][INternos]# \t #[Multiply the inner terms.][Multiplica los términos internos]# \t $\color{#aaaaaa}{(a + \bold{\color{#026fc1}{b}})(\bold{\color{#026fc1}{c}} + d)}$ \t $\color{#026fc1}{b \times c = bc}$ \\ \t #[L][UL]# \t #[Last][ÚLtimos]# \t #[Multiply the last terms.][Multiplica los términos últimos]# \t $\color{#aaaaaa}{(a + \bold{\color{#de6c00}{b}})(c + \bold{\color{#de6c00}{d}})}$ \t $\color{#de6c00}{b \times d = bd}$

Then add them all up

#[Result][Resultado]# \gap[10] \t $(a+b)(c+d) = \color{#c1026f}{ac} + \color{#0ea05e}{ad} + \color{#026fc1}{bc} + \color{#de6c00}{bd}$ \t #[F][P]# #[O][EX]# #[I][IN]# #[L][UL]#

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Examples

$(x + 1)(3x - 2)$ \t ${}=\color{#c1026f}{x \cdot (3x)} + \color{#0ea05e}{x \cdot (-2)} + \color{#026fc1}{1 \cdot (3x)} + \color{#de6c00}{1 \cdot (-2)}$ \t \gap[20] FOIL \\ \t ${}=3x^2 -2x + 3x -2$ \\ \t ${}=3x^2 +x -2$ \\ \t \\ $(x - 3)(x + 3)$ \t ${}=\color{#c1026f}{x \cdot (x)} + \color{#0ea05e}{x \cdot 3} + \color{#026fc1}{(-3) \cdot x} + \color{#de6c00}{(-3) \cdot 3}$ \t \gap[20] FOIL \\ \t ${}=x^2 + 3x - 3x - 9$ \\ \t ${}=x^2 - 9$

Special cases

The second example above, and some you filled in, are important enough to warrant special mention.

#[Some identities][Algunas identidades]#

#[If $a$ and $b$ are any real numbers, then:][Si $a$ y $b$ son cualquieras números reales, entonces:]#

#[Identity][Identidad]#	Examples
#[Difference of two squares][Diferencia de dos cuadrados]#: $(a + b)(a - b)$ \t ${}=a^2-b^2$ \\ $(a - b)(a + b)$ \t ${}=a^2-b^2$ (#[See the second example of %%FOIL above.][Ve el segundo ejemplo de %%FOIL de arriba.]#)	$(x-3)(x+3)$ \t ${}=x^2 - 3^2$ \\ \t ${}= x^2 - 9$ $(3x + 4y)(3x - 4y)$ \t ${}=(3x)^2 - (4y)^2$ \\ \t ${}=9x^2 - 16y^2$
#[Square of a sum or difference][Cuadrado de una suma o diferencia]#: $(a + b)^2$ \t ${}=a^2 + 2ab + b^2$ \\ $(a - b)^2$ \t ${}=a^2 - 2ab + b^2$ (#[See the second question above.][Ve la segunda pregunta de arriba.]#)	$(x + 3)^2$ \t ${}=x^2 + 2(x)(3) + 3^2$ \\ \t ${}=x^2 + 6x + 9$ \\ $(3x + 4y)^2$ \t ${}= (3x)^2 + 2(3x)(4y) + (4y)^2$ \\ \t ${}= 9x^2 + 24xy + 16y^2$ \\ $(3x - 4y)^2$ \t ${}=(3x)^2 - 2(3x)(4y) + (4y)^2$ \\ \t ${}=9x^2 - 24xy + 16y^2$

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Now try the exercises in Section 0.4 in Finite Mathematics and Applied Calculus. or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.