Solving Equations

1.2 Expressions and Factoring

Terms and Expressions

Before we solve equations, which is our goal for the first part of this class, we need to get fluent at reading algebraic expressions. Think of an expression like a sentence: each term is a word, and the operations tell us how those words are connected.

Definition: Terms and Expressions

A term is a number, a variable, or a product of numbers and variables. Terms are separated by addition or subtraction signs. A collection of terms is called an algebraic expression.

Example: Read the structure of an expression

Identify the terms, coefficients, and constant in

$$4x-7y+5.$$

Show Solution

Rewrite subtraction as addition:

$$4x+(-7y)+5.$$

Now we can read parts clearly:

  1. Terms: $4x$, $-7y$, and $5$.
  2. Coefficients: $4$ (for $x$) and $-7$ (for $y$).
  3. Constant term: $5$.

Definition: Like Terms

Two terms are like terms when they have exactly the same variable part (same variables raised to the same powers). Only like terms can be combined by adding or subtracting coefficients.

Example: Decide whether terms are like terms

For each pair, decide if the terms are like terms.

  1. $3x^2$ and $5x^2$
  2. $-2xy^2$ and $3x^2y$
Show Solution
  1. Yes. Both have variable part $x^2$, so they are like terms.
  2. No. One has variable part $xy^2$ and the other has $x^2y$, so they are not like terms.
MyOpenMath: Combine like terms

Expanding and Simplifying Expressions

In this section we'll be doing the same thing, but two different directions. We will simplify expressions and factor them. When students say "simplify," they usually mean combine like terms. When they say "expand," they usually mean distribute multiplication and remove parentheses first, then simplify.

Fact: Expand then combine

A reliable order is:

  1. Distribute multiplication over parentheses.
  2. Group like terms.
  3. Add or subtract coefficients.

Example: Expand and simplify a multistep expression

Simplify

$$4x(5y-3)+12x-8xy.$$

Show Solution

Distribute first:

$$4x(5y-3)+12x-8xy=20xy-12x+12x-8xy.$$

Group like terms:

$$(20xy-8xy)+(-12x+12x).$$

Combine coefficients:

$$12xy+0x=12xy.$$

So the simplified result is

$$12xy. $$

MyOpenMath: Expand and simplify

Example: Expand and simplify binomials

Simplify

$$(2x+3y)(x-2y).$$

Show Solution

Use distributive multiplication term by term:

$$ (2x+3y)(x-2y)=2x\cdot x+2x\cdot(-2y)+3y\cdot x+3y\cdot(-2y). $$

So

$$2x^2-4xy+3xy-6y^2=2x^2-xy-6y^2.$$

Final answer:

$$2x^2-xy-6y^2. $$

MyOpenMath: Expand and simplify

Theorem: Common Expansion Rules

For real numbers $a$ and $b$,

  1. $(a+b)^2=a^2+2ab+b^2$
  2. $(a-b)^2=a^2-2ab+b^2$
  3. $(a+b)(a-b)=a^2-b^2$

Example: Apply a square formula

Simplify

$$(2x-3y)^2.$$

Show Solution

Apply $(a-b)^2=a^2-2ab+b^2$ with $a=2x$ and $b=3y$:

$$ (2x-3y)^2=(2x)^2-2(2x)(3y)+(3y)^2 =4x^2-12xy+9y^2. $$

So

$$(2x-3y)^2=4x^2-12xy+9y^2. $$

MyOpenMath: Expand and simplify

Factoring Expressions

Factoring is the reverse of expansion. We look for structure and rewrite an expression as a product. This is one of the most useful skills to build in algebra, but also the most difficult.

Definition: Greatest Common Factor (GCF)

The greatest common factor of several terms is the largest factor (numeric and variable) that divides each term.

Example: Factor out the GCF

Factor completely:

$$2x^2-4x.$$

Show Solution

Find the common factor in both terms: each has $2x$.

Rewrite each term to expose it:

$$2x^2-4x=2x\cdot x-2x\cdot 2.$$

Now factor out $2x$:

$$2x^2-4x=2x(x-2).$$

So the factored form is

$$2x(x-2). $$

MyOpenMath: Factor with GCF

Example: Factor another expression by GCF

Factor completely:

$$24x^2y-36xy^3.$$

Show Solution

Numeric GCF of $24$ and $36$ is $12$.

Variable GCF: each term has $x$ and $y$, so variable GCF is $xy$.

Factor out $12xy$:

$$24x^2y-36xy^3=12xy(2x)-12xy(3y^2)=12xy(2x-3y^2).$$

Final answer:

$$12xy(2x-3y^2). $$