4.4 Factoring and Quadratic Basics

Factoring in Algebra

Factoring is the process of breaking down an algebraic expression into a product of simpler expressions or factors. It is the inverse operation of distributing or multiplying. On the SHSAT, factoring is a critical skill for simplifying algebraic fractions, finding solutions to quadratic equations, and solving complex expression puzzles. Master these four fundamental factoring methods to excel on test day.

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Greatest Common Factor (GCF) Factoring

The first step in any factoring problem is always to look for the Greatest Common Factor (GCF). The GCF is the largest number and variable term that divides evenly into every term in the expression.

Factoring Out the GCF

1. Identify the greatest common numerical factor of all coefficients.
2. Identify the lowest exponent for each common variable.
3. Divide each term by the GCF and write the GCF outside parentheses, with the remaining quotients inside.

• Example: Factor completely: $6x^3 - 15x^2$.
  • Step 1: Find the GCF of the coefficients $6$ and $15$. The largest integer that divides both is $3$.
  • Step 2: Find the GCF of the variable parts $x^3$ and $x^2$. The lowest exponent is $2$, so the variable GCF is $x^2$.
  • Step 3: Combine them to get a total GCF of $3x^2$.
  • Step 4: Divide each term of the original expression by $3x^2$: $\frac{6x^3}{3x^2} = 2x$ $\frac{-15x^2}{3x^2} = -5$
  • Step 5: Write the factored form: $3x^2(2x - 5)$ You can check this by distributing the $3x^2$ back into the parentheses.

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Factoring the Difference of Squares

A difference of squares is a binomial (two terms) where both terms are perfect squares and they are separated by a subtraction sign. The formula for factoring a difference of squares is: $a^2 - b^2 = (a - b)(a + b)$

Identifying Perfect Squares

You should memorize the first 15 perfect squares ($1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225$). A variable term is a perfect square if its exponent is even (e.g., $x^2, y^4, z^6$).

• Example: Factor $9x^2 - 16$.
  • Step 1: Confirm it is a difference of squares. $9x^2$ is $(3x)^2$ and $16$ is $4^2$. They are separated by subtraction.
  • Step 2: Using the formula with $a = 3x$ and $b = 4$, write the factors: $(3x - 4)(3x + 4)$
• Traps to Avoid:
  • Sum of Squares: The expression $a^2 + b^2$ (with a plus sign) is NOT factorable using real numbers. Do not attempt to write it as $(a+b)(a+b)$ or $(a+b)(a-b)$.
  • Multi-Step Factoring: Always check for a GCF first. For example, $2x^2 - 50$ is not immediately a difference of squares because $2$ and $50$ are not perfect squares. However, factoring out the GCF $2$ yields: $2(x^2 - 25)$ Now, the term inside is a difference of squares: $2(x - 5)(x + 5)$

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Factoring Trinomials ($x^2 + bx + c$)

A quadratic trinomial is an expression in the form $x^2 + bx + c$, where $b$ and $c$ are constant coefficients. To factor these, you must find two numbers that multiply to $c$ and add to $b$.

The Sign Rules for Trinomials

Understanding the relationship between the signs of $b$ and $c$ is a massive time-saver.

• If $c$ is positive: The two numbers must have the same sign.

  • If $b$ is positive, both numbers are positive. (e.g., $x^2 + 5x + 6 = (x + 2)(x + 3)$)
  • If $b$ is negative, both numbers are negative. (e.g., $x^2 - 5x + 6 = (x - 2)(x - 3)$)

• If $c$ is negative: The two numbers must have opposite signs. The number with the larger absolute value will match the sign of $b$.

• Example: Factor $x^2 - 5x - 6$. We need two numbers that multiply to $-6$ and add to $-5$.

  • Listing factor pairs of $-6$:
    • $1 \text{ and } -6 \Rightarrow 1 + (-6) = -5$ (This matches!)
    • $-1 \text{ and } 6 \Rightarrow -1 + 6 = 5$
    • $2 \text{ and } -3 \Rightarrow 2 + (-3) = -1$
    • $-2 \text{ and } 3 \Rightarrow -2 + 3 = 1$ The correct numbers are $1$ and $-6$. The factored form is: $(x - 6)(x + 1)$ A common error is choosing $-2$ and $-3$. While they add to $-5$, they multiply to $+6$, not $-6$.

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Solving Simple Quadratic Equations

A quadratic equation is an equation containing a squared variable, written in the standard form $ax^2 + bx + c = 0$. Solving a quadratic equation typically yields two solutions. The most reliable method on the SHSAT is factoring and using the Zero Product Property.

The Zero Product Property

The Zero Product Property states that if the product of two quantities is zero, then at least one of the quantities must be zero: $\text{If } A \times B = 0, \text{ then } A = 0 \text{ or } B = 0$

Step-by-Step Solving by Factoring

1. Set to zero: Move all terms to one side of the equation so that the other side equals $0$.
2. Factor completely using the methods described above.
3. Set each factor to zero and solve the resulting linear equations.

• Example: Solve $x^2 - x - 12 = 0$.
  • Step 1: The equation is already set to $0$.
  • Step 2: Factor the trinomial. We need numbers that multiply to $-12$ and add to $-1$. These are $-4$ and $3$. $(x - 4)(x + 3) = 0$
  • Step 3: Set each factor to $0$: $x - 4 = 0 \Rightarrow x = 4$ $x + 3 = 0 \Rightarrow x = -3$ The solutions are $x = 4$ and $x = -3$.