FOIL and multiplying polynomials

FOIL-ing or each by each

When you multiply polynomials, you can use the FOIL method or the each-by-each method. They’re the same idea: multiply every term in one polynomial by every term in the other polynomial, then combine like terms.

FOIL is a memory aid specifically for multiplying two binomials.

F - First (multiply the first terms within each parentheses)
O - Outside (multiply the two terms on the farthest edge of the parentheses)
I - Inside (multiply the two terms closest to the inside)
L - Last (multiply the last terms within each parentheses)

The FOIL method on (a + b)(c + d): arrows join the First terms a and c, the Outside terms a and d, the Inside terms b and c, and the Last terms b and d, giving ac + ad + bc + bd. — FOIL method graphic

Example 1:

Distribute $(b + 2) (b + 3)$ .

Use FOIL:

$(b + 2) (b + 3) = b^{2} + 3 b + 2 b + 6 = b^{2} + 5 b + 6$

Watch out for squaring a binomial: $(x + 3)^{2}$ means $(x + 3) (x + 3) = x^{2} + 6 x + 9$ , not $x^{2} + 9$ - FOIL still gives you a middle term.

Example 2:

Multiply $(y + 7) (y^{2} + 2 y + 3)$ .

Here you’re multiplying a binomial by a trinomial, so use the same each-by-each idea: multiply each term in $(y + 7)$ by each term in $(y^{2} + 2 y + 3)$ , then combine like terms.

$(y + 7) (y^{2} + 2 y + 3) = y (y^{2} + 2 y + 3) + 7 (y^{2} + 2 y + 3) = y^{3} + 2 y^{2} + 3 y + 7 y^{2} + 14 y + 21 = y^{3} + 9 y^{2} + 17 y + 21$

Steps taken:

Multiply each term in one expression by each term in the other
Combine like terms

Knowledge check: Expand

$(x + 4) (x + 5)$

(spoiler)

$x^{2} + 9 x + 20$

Greatest common factor

Definitions

Greatest common factor: The largest shared factor between a set of terms
Factor: A value that divides evenly into another value.

We can use factors to simplify an expression with variables or multiple terms.

Let’s say we have the polynomial $12 x^{3} + 8 x^{2}$ and we want to find the common factors between the two terms. We’ll want to look at values that go into both the terms $12$ and $8$ and both of the variables $x^{3}$ and $x^{2}$ . If we’re looking specifically for the greatest common factor, we’ll want to find the largest values that go into each term.

The factors of $12$ are $1, 2, 3, 4, 6,$ and $12$

The factors of $8$ are $1, 2, 4,$ and $8$

They share factors of $1, 2,$ and $4$ , with $4$ being the greatest common factor.

Now look at the variables: take the variable to the smallest exponent that appears across the terms. Both $12 x^{3}$ and $8 x^{2}$ include powers of $x$ , and the smallest exponent is $2$ , so $x^{2}$ is the shared factor.

So the greatest common factor between $12 x^{3}$ and $8 x^{2}$ is $4 x^{2}$ .

We can now rewrite the expression by factoring out $4 x^{2}$ (divide each term by $4 x^{2}$ ):

$12 x^{3} + 8 x^{2} = 4 x^{2} (3 x + 2)$

Steps taken:

Find the GCF of the coefficients
Find the GCF of the variables
Multiply to get the overall GCF
Factor it out of each term

Factoring

Definitions

Factoring: Rewriting an expression as a product of simpler factors. It is the reverse of expanding (FOIL-ing).

Factoring is often called reverse FOIL-ing: instead of starting with two parentheses and multiplying them into one expression, we start with the expanded expression and work backward to find the two parentheses that created it.

For example, FOIL-ing $(x + 3) (x - 5)$ gives:

$(x + 3) (x - 5) = x^{2} - 5 x + 3 x - 15 = x^{2} - 2 x - 15$

Factoring starts with $x^{2} - 2 x - 15$ and asks: “what two parentheses could multiply together to create this expression?”

How do we factor?

We will work with expressions in the form:

$a x^{2} + b x + c$

For:

$x^{2} - 2 x - 15$

The values are:

$a = 1$
$b = - 2$
$c = - 15$

When $a = 1$ , as here, we look for two numbers that:

Multiply to give $c$
Add to give $b$

So our factors will look like:

$(x \pm_) (x \pm_)$

At first, we do not know which numbers belong in the parentheses, so we list the factor pairs of $- 15$ :

$(- 1, 15)$

$(1, - 15)$

$(- 3, 5)$

$(3, - 5)$

Since $c$ is negative, one number must be positive and one must be negative.

Now check which pair adds to $- 2$ :

$- 1 + 15 1 + (- 15) - 3 + 5 3 + (- 5) = 14 = - 14 = 2 = - 2$

The pair that works is:

$3 and - 5$

So the correct factors are:

$(x + 3) (x - 5)$

Check your answer:

FOIL-ing $(x + 3) (x - 5)$ gets you back to $x^{2} - 2 x - 15$ - the same expansion shown at the top of this section. If the expression you get after FOIL-ing matches the original problem, the factoring is correct.

Pitfall: a factor pair has to satisfy both conditions - it must multiply to $c$ and add to $b$ . Check the sign before you commit to a pair: if $c$ is positive, both numbers share the sign of $b$ ; if $c$ is negative, the numbers have opposite signs.

Example: leading coefficient other than 1

Factor $2 x^{2} + 7 x + 3$ .

Here $a = 2$ , and $2 x^{2}$ splits only as $2 x \cdot x$ , so the factors take the form $(2 x \pm_) (x \pm_)$ . (When $a$ has more than one factor pair, test each split: $6 x^{2}$ could start $(6 x \pm_) (x \pm_)$ or $(3 x \pm_) (2 x \pm_)$ .) Since $c = 3$ is positive and $b = 7$ is positive, both numbers are positive. Testing the factor pair of $3$ (just $1$ and $3$ ), the outer and inner products need to add to $7 x$ :

$(2 x + 1) (x + 3) = 2 x^{2} + 6 x + x + 3 = 2 x^{2} + 7 x + 3$

Answer: $(2 x + 1) (x + 3)$

Example: difference of squares

Factor $x^{2} - 9$ .

There’s no middle term, which means $b = 0$ : the two numbers must add to $0$ , so they’re opposites that multiply to $- 9$ - that’s $3$ and $- 3$ .

Answer: $(x + 3) (x - 3)$

Solving by factoring

If we can factor a quadratic equation, we can solve for $x$ by setting each factor equal to zero.

This works because when two expressions multiply to equal $0$ , at least one of them must equal $0$ .

Example:

$x^{2} - 2 x - 15 (x + 3) (x - 5) = 0 = 0$

Now set each factor equal to $0$ :

$x + 3 x = 0 = - 3 x - 5 x = 0 = 5$

So the solutions are:

$x = - 3 and x = 5$

Each solution has the opposite sign of the number in its factor: $x + 3$ gives $- 3$ , and $x - 5$ gives $5$ .

Steps taken:

Factor the quadratic expression
Set each factor equal to zero
Solve each resulting equation
State the solutions

Knowledge check: Solve

$x^{2} + 5 x + 6 = 0$

(spoiler)

$x = - 2 and x = - 3$

FOIL-ing or each by each

FOIL is a memory aid specifically for multiplying two binomials.

Example 1:

Distribute $(b + 2) (b + 3)$ .

Use FOIL:

$(b + 2) (b + 3) = b^{2} + 3 b + 2 b + 6 = b^{2} + 5 b + 6$

Watch out for squaring a binomial: $(x + 3)^{2}$ means $(x + 3) (x + 3) = x^{2} + 6 x + 9$ , not $x^{2} + 9$ - FOIL still gives you a middle term.

Example 2:

Multiply $(y + 7) (y^{2} + 2 y + 3)$ .

Here you’re multiplying a binomial by a trinomial, so use the same each-by-each idea: multiply each term in $(y + 7)$ by each term in $(y^{2} + 2 y + 3)$ , then combine like terms.

$(y + 7) (y^{2} + 2 y + 3) = y (y^{2} + 2 y + 3) + 7 (y^{2} + 2 y + 3) = y^{3} + 2 y^{2} + 3 y + 7 y^{2} + 14 y + 21 = y^{3} + 9 y^{2} + 17 y + 21$

Steps taken:

Multiply each term in one expression by each term in the other
Combine like terms

Knowledge check: Expand

$(x + 4) (x + 5)$

(spoiler)

$x^{2} + 9 x + 20$

Greatest common factor

Definitions

Greatest common factor: The largest shared factor between a set of terms
Factor: A value that divides evenly into another value.

We can use factors to simplify an expression with variables or multiple terms.

Let’s say we have the polynomial $12 x^{3} + 8 x^{2}$ and we want to find the common factors between the two terms. We’ll want to look at values that go into both the terms $12$ and $8$ and both of the variables $x^{3}$ and $x^{2}$ . If we’re looking specifically for the greatest common factor, we’ll want to find the largest values that go into each term.

The factors of $12$ are $1, 2, 3, 4, 6,$ and $12$

The factors of $8$ are $1, 2, 4,$ and $8$

They share factors of $1, 2,$ and $4$ , with $4$ being the greatest common factor.

Now look at the variables: take the variable to the smallest exponent that appears across the terms. Both $12 x^{3}$ and $8 x^{2}$ include powers of $x$ , and the smallest exponent is $2$ , so $x^{2}$ is the shared factor.

So the greatest common factor between $12 x^{3}$ and $8 x^{2}$ is $4 x^{2}$ .

We can now rewrite the expression by factoring out $4 x^{2}$ (divide each term by $4 x^{2}$ ):

$12 x^{3} + 8 x^{2} = 4 x^{2} (3 x + 2)$

Steps taken:

Find the GCF of the coefficients
Find the GCF of the variables
Multiply to get the overall GCF
Factor it out of each term

Factoring

Definitions

Factoring: Rewriting an expression as a product of simpler factors. It is the reverse of expanding (FOIL-ing).

For example, FOIL-ing $(x + 3) (x - 5)$ gives:

$(x + 3) (x - 5) = x^{2} - 5 x + 3 x - 15 = x^{2} - 2 x - 15$

Factoring starts with $x^{2} - 2 x - 15$ and asks: “what two parentheses could multiply together to create this expression?”

How do we factor?

We will work with expressions in the form:

$a x^{2} + b x + c$

For:

$x^{2} - 2 x - 15$

The values are:

$a = 1$
$b = - 2$
$c = - 15$

When $a = 1$ , as here, we look for two numbers that:

Multiply to give $c$
Add to give $b$

So our factors will look like:

$(x \pm_) (x \pm_)$

At first, we do not know which numbers belong in the parentheses, so we list the factor pairs of $- 15$ :

$(- 1, 15)$

$(1, - 15)$

$(- 3, 5)$

$(3, - 5)$

Since $c$ is negative, one number must be positive and one must be negative.

Now check which pair adds to $- 2$ :

$- 1 + 15 1 + (- 15) - 3 + 5 3 + (- 5) = 14 = - 14 = 2 = - 2$

The pair that works is:

$3 and - 5$

So the correct factors are:

$(x + 3) (x - 5)$

Check your answer:

Example: leading coefficient other than 1

Factor $2 x^{2} + 7 x + 3$ .

Here $a = 2$ , and $2 x^{2}$ splits only as $2 x \cdot x$ , so the factors take the form $(2 x \pm_) (x \pm_)$ . (When $a$ has more than one factor pair, test each split: $6 x^{2}$ could start $(6 x \pm_) (x \pm_)$ or $(3 x \pm_) (2 x \pm_)$ .) Since $c = 3$ is positive and $b = 7$ is positive, both numbers are positive. Testing the factor pair of $3$ (just $1$ and $3$ ), the outer and inner products need to add to $7 x$ :

$(2 x + 1) (x + 3) = 2 x^{2} + 6 x + x + 3 = 2 x^{2} + 7 x + 3$

Answer: $(2 x + 1) (x + 3)$

Example: difference of squares

Factor $x^{2} - 9$ .

There’s no middle term, which means $b = 0$ : the two numbers must add to $0$ , so they’re opposites that multiply to $- 9$ - that’s $3$ and $- 3$ .

Answer: $(x + 3) (x - 3)$

Solving by factoring

If we can factor a quadratic equation, we can solve for $x$ by setting each factor equal to zero.

This works because when two expressions multiply to equal $0$ , at least one of them must equal $0$ .

Example:

$x^{2} - 2 x - 15 (x + 3) (x - 5) = 0 = 0$

Now set each factor equal to $0$ :

$x + 3 x = 0 = - 3 x - 5 x = 0 = 5$

So the solutions are:

$x = - 3 and x = 5$

Each solution has the opposite sign of the number in its factor: $x + 3$ gives $- 3$ , and $x - 5$ gives $5$ .

Steps taken:

Factor the quadratic expression
Set each factor equal to zero
Solve each resulting equation
State the solutions

Knowledge check: Solve

$x^{2} + 5 x + 6 = 0$

(spoiler)

$x = - 2 and x = - 3$