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

(This topic is also in Section 0.3 in Finite Mathematics and Applied Calculus)

%%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]# ${}= \dfrac{1}{2} + \dfrac{2z}{3x^2y}$

Suggested video for this topic: Video by Papapodcasts

* #[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]#

Suggested video for this topic: Video by MATHRoberg

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 ${}=9x^2 - 24xy + 16y^2$

Suggested video for this topic: Video by Math and Science

Now try the exercises in Section 0.3 in Finite Mathematics and Applied Calculus. or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.