In this section, you’ll practice simplifying, expanding, factoring, and rewriting algebraic expressions using the commutative, associative, and distributive properties. These skills help you create equivalent forms that are easier to evaluate or use in solving.

Definitions

Term

A single algebraic component made up of a coefficient and variables multiplied together, for example $3x^2$ or $-5y$.

Coefficient

The numerical factor of a term, such as $3$ in $3x^2$.

Like terms

Terms that have the same variables raised to the same powers, which can be combined by adding or subtracting coefficients.

Distributive property

The rule $a(b+c)=ab+ac$, which allows expansion or factoring of expressions.

Equivalent expressions

Different forms of an expression that simplify to the same value for all variable inputs.

Quadratic trinomial

An expression of the form $x^2+bx+c$ where the coefficient of $x^2$ is $1$.

Expression

Combines numbers, variables, and operations to represent a value (e.g. $3x-2$).

Equation

Sets two expressions equal, often containing an unknown to solve for (e.g. $3x-2=7$).

• An expression has no $=$ sign, so it can’t be “solved.” You can only simplify it or evaluate it for given values.
• An equation includes $=$ and asks for the value(s) of the variable that make the two sides equal.
• Every equation contains two expressions (one on each side), but an expression doesn’t become an equation unless you set it equal to something.

Simplifying by combining like terms

Simplifying an expression means rewriting it in a cleaner form by combining like terms. Like terms have identical variable parts (including exponents). For example, $3x^2$ and $-5x^2$ are like terms because both include $x^2$. But $3x^2$ and $3x$ are not like terms because the exponents on $x$ are different.

Why combine like terms?

• Clarity: A simplified expression is easier to read and interpret.
• Efficiency: Fewer terms means less work when you substitute values.
• Preparation: Many techniques (like factoring and solving equations) assume the expression is already simplified.

How to identify and combine like terms:

• Scan for matching variable parts: Look for terms with exactly the same variables raised to the same powers (the coefficients can differ).
• Group them together: Use the commutative and associative properties to rearrange terms so like terms are next to each other.
• Add or subtract coefficients: Combine only the numbers in front.
• Keep the variable part: Attach the common variable part to the new coefficient.

Example: Combining like terms Given the expression

$$2x+5x-3x+4y-y$$

• Identify like terms:
  • $x$-terms: $2x,5x,-3x$
  • $y$-terms: $4y,-y$
• Combine the $x$-terms:
  • Coefficients: $2+5-3=4$
  • Result: $4x$
• Combine the $y$-terms:
  • Coefficients: $4-1=3$
  • Result: $3y$
• Final simplified form:

$$4x+3y$$

Answer: $4x+3y$

Example: Simplify with multiple variables

$$3a+2b-5a+4b$$

(spoiler)

• Identify like terms $3a,-5a$ and $2b,4b$
• Combine $a$ terms: $3-5=-2$ so $-2a$
• Combine $b$ terms: $2+4=6$ so $6b$
• Result $-2a+6b$

Answer: $-2a+6b$

Expanding using the distributive property

The distributive property lets you remove parentheses by multiplying a factor across each term inside the parentheses. In symbols:

$$a(b+c)=ab+ac.$$

Why expand?

• Simplification: Removing parentheses can make it easier to combine like terms.
• Solving: Many algebra steps work best when expressions aren’t nested in parentheses.
• Clarity: An expanded form shows every product explicitly.

How to expand any expression

• Identify the factor (or expression) directly outside the parentheses.
• Multiply it by each term inside the parentheses.
• Write each product, keeping track of signs.
• Combine like terms if any appear.

Step-by-step

• Given $k(T_1+T_2+\cdots +T_n)$
  1. Multiply $k\cdot T_1$
  2. Multiply $k\cdot T_2$
  3. …
  4. Multiply $k\cdot T_n$
• Write the sum of these products:

  $$k{T}_1+k{T}_2+\cdots +k{T}_n.$$

Example: Expand $4(x+3)$

• Distribute $4$ to each term: $4x$, $12$
• Result $4x+12$

Answer: $4x+12$

Example: Expand $5(2x-7)$

(spoiler)

• Distribute $5$ to each term:
  • $5\cdot 2x=10x$
  • $5\cdot (-7)=-35$
• Result $10x-35$

Answer: $10x-35$

Example: Expand a binomial product $(x+2)(x-5)$ When multiplying two binomials, FOIL - First, Outer, Inner, Last - organizes the distributive steps:

• First: multiply the first terms in each binomial

  $$x\cdot x=x^2$$

• Outer: multiply the outer terms

  $$x\cdot (-5)=-5x$$

• Inner: multiply the inner terms

  $$2\cdot x=2x$$

• Last: multiply the last terms in each binomial

  $$2\cdot (-5)=-10$$

• Combine like terms: add the Outer and Inner results

  $$-5x+2x=-3x$$

Result:

$$>x^2-3x-10$$

Answer: $x^2-3x-10$

Example: Expand $(2x+3)(x-4)$

(spoiler)

• First: $2x\cdot x=2x^2$
• Outer: $2x\cdot (-4)=-8x$
• Inner: $3\cdot x=3x$
• Last: $3\cdot (-4)=-12$
• Combine like terms: $-8x+3x=-5x$

Result:

$$>2x^2-5x-12$$

Answer: $2x^2-5x-12$

Simplifying more complex expressions

Example: Simplify $2(x+1)-3(x-2)$

(spoiler)

• Distribute $2$: $2x+2$
• Distribute $-3$: $-3x+6$
• Combine like terms: $(2x-3x)+(2+6)=-x+8$
• Result $-x+8$

Answer: $-x+8$

Example: Simplify $\frac{2x^2}{6x}$

(spoiler)

• Factor coefficients and variables: $\frac{2}{6}\cdot \frac{x^2}{x}$
• Simplify $\frac{2}{6}=\frac{1}{3}$ and $\frac{x^2}{x}=x$
• Result $\frac{1}{3}x$

Answer: $\frac{1}{3}x$

Factoring expressions

Factoring reverses expansion. Instead of distributing a factor across terms, you rewrite an expression as a product of factors by pulling out common factors or using known patterns. A factored form is often easier to evaluate, simplify further, or use when solving equations.

• Greatest common factor (GCF): the largest factor shared by all terms; factoring it out simplifies the expression.
• Structure recognition: look for patterns such as differences of squares or quadratic trinomials.

Sidenote

Reasons to factor

• Reveals roots of related equations
• Simplifies evaluation and algebraic manipulation
• Prepares expressions for solving equations or finding common denominators

Example: Factor out a common factor Factor

$$6x+9$$

• List factors of each term:
  • $6x$: factors are $1\cdot 6x$, $2\cdot 3x$, $3\cdot 2x$, $6\cdot x$
  • $9$: factors are $1\cdot 9$, $3\cdot 3$, $9\cdot 1$
• Identify the GCF of all terms: both share a factor of $3$.
• Divide each term by $3$ to find the co-factor:
  • $6x÷3=2x$
  • $9÷3=3$
• Write the factored form:

$$>6x+9=3(2x+3).$$

Answer: $3(2x+3)$

Rewriting expressions

Rewriting means using expansion or factoring to create an equivalent form that’s more useful for a particular goal - evaluation, simplification, or solving.

Example: Rewrite as a product (difference of squares) Rewrite

$$>x^2-9$$

• Recognize $9=3^2$, so $x^2-9=x^2-3^2$ fits the pattern $a^2-b^2=(a-b)(a+b)$.
• Identify $a=x$, $b=3$.
• Apply the formula:

$$>x^2-9=(x-3)(x+3).$$

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

Example: Rewrite to factor a common binomial Rewrite

$$4(x-1)+2(x-1)$$

• Notice both terms contain the factor $(x-1)$.
• Factor out the common binomial:

$$>4(x-1)+2(x-1)=(x-1)(4+2).$$

• Simplify inside the parentheses:

$$=6(x-1).$$

Answer: $6(x-1)$

Factoring quadratics with leading coefficient one

When you have a quadratic of the form

$$x^2+bx+c$$

you can factor it in one of two equivalent ways:

1. Direct product-sum method

   • Find two numbers $m$ and $n$ such that

     $$m+n=b\text{and}m\cdot n=c.$$

   • Then

     $$x^2+bx+c=(x+m)(x+n).$$

2. Factoring by grouping

   • Rewrite the middle term $bx$ as $(mx+nx)$:

     $$x^2+bx+c=x^2+mx+nx+c.$$

   • Group into pairs and factor each:

     $$(x^2+mx)+(nx+c)=x(x+m)+n(x+m)=(x+m)(x+n).$$

Example: Factor $(x^2-5x+6)$ Direct method:

• This is of the form $x^2+bx+c$ with $b=-5$, $c=6$.
• We seek two numbers $m,n$ such that $m+n=-5$ and $m\cdot n=6$.
• List integer pairs multiplying to $6$: $(1,6),(2,3),(-1,-6),(-2,-3)$.
• Find the pair summing to $-5$: $-2+(-3)=-5$.
• Since $-2\cdot (-3)=6$ and $-2+(-3)=-5$, we have our factors.
• Rewrite $-5x$ as $-2x-3x$ and factor by grouping (or apply the direct pattern). Both methods are shown below:

$$>x^2-5x+6=(x-2)(x-3).$$

• Grouping method:
  • Rewrite $-5x$ as $(-2x-3x)$

$$>x^2-5x+6=x^2-2x-3x+6.$$

• Group:

$$>(x^2-2x)+(-3x+6).$$

• Factor each group:

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

• Factor the common binomial:

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

Answer: $(x-2)(x-3)$

Example: Factor $(x^2+7x+12)$

(spoiler)

• Direct method: find $(m,n)$ such that

$$>m+n=7,m\cdot n=12.$$

The pair $m=3$ and $n=4$ works, giving

$$>x^2+7x+12=(x+3)(x+4).$$

• Grouping method: rewrite $(7x)$ as $(3x+4x)$:

$$>x^2+7x+12=x^2+3x+4x+12=(x^2+3x)+(4x+12)=x(x+3)+4(x+3)=(x+3)(x+4).$$

Answer: $(x+3)(x+4)$

Plugging in values for variables

When an expression contains one or more variables, you can evaluate it by substituting the given values and then simplifying using the correct order of operations: parentheses first, then multiplication/division, then addition/subtraction. Be especially careful with negative numbers.

• Identify the variables and their assigned values.
• Substitute each variable in the expression with its corresponding number.
• Simplify step by step, by following the order operations PEMDAS:
  1. Evaluate inside parentheses and apply exponents if applicable.
  2. Perform multiplications and divisions.
  3. Perform additions and subtractions.
• Check your result by verifying one more time.

Example: Given

$$2x+3y=7,$$

find $y$ when $x=4$.

• Substitute $x=4$ into $2x+3y=7$:

$$2(4)+3y=7$$

• Simplify the multiplication:

$$8+3y=7$$

• Subtract $8$ from both sides:

$$3y=-1$$

• Divide both sides by $3$:

$$>y=-\frac{1}{3}$$

Answer: $-\frac{1}{3}$

Example: Evaluate

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

when $x=2$ and $y=-3$.

(spoiler)

• Substitute $x=2$ and $y=-3$:

$$>7(2)-4(2+(-3))$$

• Simplify inside the parentheses:

$$2+(-3)=-1$$

so the expression becomes

$$>7\cdot 2-4(-1)$$

• Multiply:

$$14-(-4)$$

• Subtracting a negative is adding its opposite:

$$14+4=18$$

• Final result:

$$18$$

Answer: $18$