Factoring Quadratic Expressions and Greatest Common Factor (GCF): Precalculus Study Notes

Factoring Quadratic Expressions

Introduction to Factoring Quadratics

Factoring quadratic expressions is a fundamental skill in algebra and precalculus. It involves rewriting a quadratic expression in the form $ax^2 + bx + c$ as a product of two binomials. This process is essential for solving quadratic equations, simplifying expressions, and understanding polynomial functions.

• Quadratic Expression: An expression of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$.

• Factoring: The process of expressing a polynomial as a product of its factors.

• Standard Form: $ax^2 + bx + c$

Factoring by Finding Two Numbers

To factor $x^2 + bx + c$, find two numbers that multiply to $c$ and add up to $b$. This method is often called "splitting the middle term" or the "ac method" when $a \neq 1$.

• Step 1: Identify $b$ (the coefficient of $x$) and $c$ (the constant term).

• Step 2: Find two integers $m$ and $n$ such that $m \times n = c$ and $m + n = b$.

• Step 3: Write the factors as $(x + m)(x + n)$.

Example: Factor $x^2 + 18x + 81$

• Find two numbers that multiply to $81$and add up to $18$.

• $9 \times 9 = 81$ and $9 + 9 = 18$

• So, $x^2 + 18x + 81 = (x + 9)(x + 9)$

Factoring with Negative Coefficients

When $b$ or $c$ is negative, the same process applies, but the signs of $m$ and $n$ must be considered.

• Example: Factor $x^2 - 13x + 40$

• Find two numbers that multiply to $40$and add up to$-13$.

• $-5 \times -8 = 40$ and $-5 + (-8) = -13$

• So, $x^2 - 13x + 40 = (x - 5)(x - 8)$

Factoring with Negative Constants

If $c$ is negative, one factor will be positive and one will be negative.

• Example: Factor $x^2 + 3x - 54$

• Find two numbers that multiply to $-54$ and add up to $3$.

• $9 \times -6 = -54$ and $9 + (-6) = 3$

• So, $x^2 + 3x - 54 = (x + 9)(x - 6)$

Factoring Quadratics with Leading Coefficient $a \neq 1$

When the leading coefficient $a$ is not 1, use the "ac method":

• Multiply $a \times c$.

• Find two numbers that multiply to $a \times c$ and add up to $b$.

• Rewrite the middle term using these numbers and factor by grouping.

Example: Factor $2x^2 - 13x + 40$

• $a = 2$, $b = -13$, $c = 40$

• $2 \times 40 = 80$

• Find two numbers that multiply to $80$and add up to$-13$($-5$and$-8$)

• Rewrite: $2x^2 - 5x - 8x + 40$

• Group: $(2x^2 - 5x) + (-8x + 40)$

• Factor: $x(2x - 5) - 8(x - 5)$

• Final: $(2x - 5)(x - 8)$

Factoring Practice Problems

• $a^2 + 4a + 3$

• $x^2 + 6x + 8$

• $m^2 + 9m + 18$

• $a^2 - 11a + 30$

• $x^2 - 7x + 12$

• $x^2 + 4x - 12$

• $x^2 - 3x - 54$

• $x^2 - 2x - 35$

For each, find two numbers that multiply to $c$ and add up to $b$.

Greatest Common Factor (GCF) and Factoring

Introduction to GCF

The Greatest Common Factor (GCF) of a polynomial is the largest expression that divides each term of the polynomial. Factoring out the GCF is often the first step in simplifying or factoring polynomials.

• Step 1: Identify the GCF of all terms.

• Step 2: Factor out the GCF from the polynomial.

• Step 3: Factor the remaining expression if possible.

Examples of GCF Factoring

• Example 1: $5x^2 + 75x + 270$ GCF: $5$$5x^2 + 75x + 270 = 5(x^2 + 15x + 54)$

• Example 2: $-4x^2 - 44x + 104$ GCF: $-4$ $-4x^2 - 44x + 104 = -4(x^2 + 11x - 26)$

• Example 3: $5x^3 + 55x^2 + 120x$ GCF: $5x$ $5x^3 + 55x^2 + 120x = 5x(x^2 + 11x + 24)$

• Example 4: $2x^2 + 24x + 22$ GCF: $2$$2x^2 + 24x + 22 = 2(x^2 + 12x + 11)$

Factoring Quadratic Expressions: Summary Table

The following table summarizes the process of factoring quadratics by identifying two numbers that multiply to $c$ and add up to $b$.

Quadratic Expression	Multiply to	Add up to	Factors
$x^2 + 18x + 81$	81	18	$(x + 9)(x + 9)$
$x^2 - 13x + 40$	40	-13	$(x - 5)(x - 8)$
$x^2 + 3x - 54$	-54	3	$(x + 9)(x - 6)$
$x^2 - 3x - 70$	-70	-3	$(x - 10)(x + 7)$
$2x^2 - 13x + 40$	80	-13	$(2x - 5)(x - 8)$

Additional info:

• Factoring quadratics is a key skill for solving equations, graphing parabolas, and simplifying rational expressions in precalculus.

• Always check for a GCF before attempting to factor quadratics.

• Factoring is also used in higher-level mathematics, such as calculus and linear algebra, for simplifying expressions and solving equations.