Simplifying and factoring linear expressions Knowledge-Based Study

Study notes

1. What is a Linear Expression?

   A linear expression is an algebraic expression in which the highest exponent of any variable is 1. For example, $3x + 5$ and $-2y + 7$ are linear expressions, whereas $x^2 + 4$ is not.

   Identifying Like Terms

   An expression is composed of terms separated by addition or subtraction signs. A term can be a constant (a number by itself) or a variable term (a product of a number, called the coefficient, and a variable).

   • Like terms: Terms that have the exact same variable raised to the same power. For example, $5x$ and $-2x$ are like terms, as are the constants $7$ and $-11$.
   • Unlike terms: Terms that have different variables or variables raised to different powers, such as $4x$ and $4y$.

   To combine like terms, add or subtract their numerical coefficients while keeping the variable unchanged:

   $6x + 4x = (6 + 4)x = 10x$

   $9a - 12a = (9 - 12)a = -3a$

   Using the Distributive Property

   Simplifying an expression means writing it in the most compact and clear form possible, with no parentheses remaining and all like terms combined. Often, this requires applying the distributive property:

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

   When a negative number precedes the parentheses, ensure the negative sign is distributed to every term inside:

   $-3(2x - 5) = -3 \cdot 2x + (-3) \cdot (-5) = -6x + 15$

   Step-by-Step Simplification Strategy

   1. Clear parentheses: Multiply through using the distributive property.
   2. Group like terms: Place variable terms together and constant terms together, taking care to keep the correct operational sign with each term.
   3. Combine coefficients: Perform addition or subtraction on the coefficients of like terms and combine constants.

   Example: Simplify $4(3x - 2) - 5x + 11$.

   Step 1: Distribute $4$:
   $12x - 8 - 5x + 11$

   Step 2: Rearrange like terms:
   $12x - 5x - 8 + 11$

   Step 3: Combine terms:
   $7x + 3$

   Factoring as the Reverse of Distribution

   Factoring is the process of rewriting an expression as a product of simpler terms or expressions. For linear expressions, factoring usually involves identifying and factoring out the greatest common factor (GCF) of all terms.

   Finding the Greatest Common Factor (GCF)

   The GCF of two or more terms is the largest quantity that divides evenly into each term.

   • For the terms in $12x + 18$, the coefficients are $12$ and $18$. The greatest common factor of $12$ and $18$ is $6$.
   • Dividing each term by $6$ gives $12x \div 6 = 2x$ and $18 \div 6 = 3$.
   • Factored form: $6(2x + 3)$.

   Factoring with Negative Coefficients

   When the leading term has a negative coefficient, it is standard practice to factor out a negative common factor so the variable term inside parentheses has a positive coefficient:

   Example: Factor $-8y - 20$.

   • The GCF of $8$ and $20$ is $4$. Factor out $-4$:
   • $-8y \div (-4) = 2y$
   • $-20 \div (-4) = 5$
   • Factored form: $-4(2y + 5)$.

   Check your work: You can always check a factored expression by redistributing to see if it yields the original expression.

   Guided Example 1: Combining Distribution and Factoring

   Task: Simplify the expression $2(5x - 1) + 4(x + 7)$, then factor the simplified result completely.

   Solution:

   • Distribute each factor: $10x - 2 + 4x + 28$
   • Combine like terms: $(10x + 4x) + (-2 + 28) = 14x + 26$
   • Find the GCF of $14x$ and $26$: The GCF is $2$.
   • Factor out $2$: $2(7x + 13)$

   Common Pitfalls to Avoid

   • Forgetting to distribute to all terms: A common error in $5(2x + 3)$ is writing $10x + 3$ instead of $10x + 15$. Remember that the outside factor applies to every term inside the parentheses.
   • Sign errors with subtraction: In $-(3x - 4)$, distributing the negative yields $-3x + 4$, not $-3x - 4$.
   • Incomplete factoring: Writing $12x + 24$ as $3(4x + 8)$ leaves a common factor of $4$ inside. Always remove the greatest common factor: $12(x + 2)$.
   • Combining unlike terms: Never combine a variable term with a constant term (for instance, $5x + 3$ cannot be simplified to $8x$).