Factoring Linear Expressions - Seventh Grade Mathematics

Factoring linear expressions reverses the distributive property, decomposing an expanded sum into a product of its greatest common factor and an equivalent binomial.

Factoring as the Inverse of Expansion

In algebra, expanding and factoring represent opposite mathematical journeys. When you expand an expression using the distributive property, you multiply a single factor into a parenthetical group to produce a sum or difference of terms. When you factor an expression, you extract a common factor from the terms to reconstruct the original product.

               EXPANSION vs FACTORING
   Expansion:   a * (b + c)   ======>   ab + ac
   Factoring:   ab + ac       ======>   a * (b + c)

Factoring reveals the underlying structural components of an expression, making it easier to evaluate, simplify fractions, and solve equations.

Extracting the Greatest Common Factor (GCF)

To factor a linear expression of the form ax plus b completely, identify the greatest common factor (GCF) of the numerical coefficients.

               FACTORING WITH THE GCF PROTOCOL
  1. Find the prime factorization of each coefficient and constant.
  2. Determine the greatest common factor (GCF) shared by all terms.
  3. Divide each term by the GCF to determine the terms remaining inside.
  4. Write the GCF outside parentheses containing the resulting quotient.

Consider factoring twelve x plus thirty: Step 1: Find prime factors: twelve equals two times two times three; thirty equals two times three times five. Step 2: Identify common prime factors: two times three equals six. The GCF is six. Step 3: Divide each term by six: twelve x divided by six equals two x; thirty divided by six equals five. Step 4: Express as a product: six times the quantity two x plus five.

Check by Distributing:
  6 * (2x + 5) = (6 * 2x) + (6 * 5) = 12x + 30

Because distribution returns the exact original expression, the factored form is guaranteed to be mathematically equivalent.

Factoring Negative and Rational Coefficients

In advanced algebra, it is often desirable to factor out a negative coefficient or a unit fraction, even when the terms do not share an obvious whole-number factor.

Factoring Out a Negative Coefficient

Factoring out a negative factor alters the signs of all terms left inside the parentheses.

Example: Factor -8x + 24
Extract -8:
  -8x / (-8) = x
  +24 / (-8) = -3
Factored Form: -8 * (x - 3)

Factoring out the negative coefficient leaves the variable x with a clean, positive coefficient of one.

Factoring Out Fractional Coefficients

When an expression contains fractional terms, factoring out a common unit fraction simplifies the terms inside the parentheses into integers.

Example: Factor (1/4)x + 3
Extract 1/4:
  [(1/4)x] / (1/4) = x
  3 / (1/4) = 3 * 4 = 12
Factored Form: (1/4) * (x + 12)

Notice that dividing three by one-fourth is equivalent to multiplying three by four, resulting in twelve. Distributing one-fourth across x plus twelve confirms that one-fourth times twelve equals three.

Interpreting Equivalent Expressions in Real-World Contexts

Rewriting an expression in factored form often clarifies the real-world meaning of the situation being modeled.

Scenario: A concert venue charges ticket price t plus a $4 service fee.
A family purchases tickets for 5 members:
Expanded Form: 5t + 20   --> Total cost broken into tickets and total fees.
Factored Form: 5 * (t + 4)--> Total cost viewed as 5 individual packages.

Both expressions calculate the identical financial total, but the factored form clearly demonstrates that each person pays ticket price t plus four dollars.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. Factor the linear expression fourteen x plus thirty-five by finding and extracting the greatest common factor.

Exercise 2. Factor the linear expression negative sixteen y plus forty by factoring out negative eight.

Exercise 3. Rewrite the expression one-third x plus five in factored form by factoring out the coefficient one-third.

Exercise 4. A school bookstore sells notebooks for n dollars each and offers a special where every notebook comes with a pen costing two dollars. For a class order of twenty-four students, write the total cost in expanded form and in factored form, explaining what each form highlights.

Solutions

Solution to Exercise 1. Find the greatest common factor of fourteen and thirty-five. The factors of fourteen are one, two, seven, and fourteen. The factors of thirty-five are one, five, seven, and thirty-five. The GCF is seven. Divide each term by seven: fourteen x divided by seven equals two x; thirty-five divided by seven equals five. The factored expression is seven times the quantity two x plus five.

Solution to Exercise 2. Divide each term of negative sixteen y plus forty by negative eight. For the first term: negative sixteen y divided by negative eight equals positive two y. For the second term: positive forty divided by negative eight equals negative five. Placing negative eight outside the parentheses yields: negative eight times the quantity two y minus five. Check: negative eight times two y is negative sixteen y, and negative eight times negative five is positive forty.

Solution to Exercise 3. Factor out one-third from both terms: one-third x divided by one-third equals x. Next, five divided by one-third is equivalent to five times three, which equals fifteen. Writing one-third outside the parentheses gives: one-third times the quantity x plus fifteen. Check: one-third times x is one-third x, and one-third times fifteen is five.

Solution to Exercise 4. The cost for one student is the notebook price plus two dollars: the quantity n plus two. In factored form, the total cost for twenty-four students is twenty-four times the quantity n plus two. This form highlights the individual cost package per student multiplied by class size. Distributing twenty-four yields the expanded form: twenty-four n plus forty-eight. This expanded form highlights the separate total costs: twenty-four n dollars spent on notebooks and forty-eight dollars spent on pens.