### Factoring with a Leading Coefficient of 1

**Factoring** is a process of writing an algebraic expression as a product. Factoring can be used with the zero product property to solve quadratic equations. The zero product property states that if the product of real numbers is zero, then one of its factors is equal to zero. In other words, if $p$ and $q$ are real numbers and $pq=0$, then $p=0$ or $q=0$.

To solve a quadratic equation by factoring:

1. Set one side of the equation equal to zero.

2. Factor the other side of the equation.

3. Set each factor equal to zero.

4. Solve the resulting equations.

To factor an expression of the form $x^2+bx+c$, look for two factors of $c$ that have a sum of $b$. When $c$ is positive, the two factors have the same sign as the sign of $b$. When $c$ is negative, the two factors have opposite signs, and the factor with the greater absolute value (greater distance from zero on a number line) will have the same sign as $b$.

Next, look for two factors of $c$, or 15, that have a sum of $b$, or 8. When $c$ is positive, the factors will have the same sign as $b$. So, the factors are $+3$ and $+5$ are positive.

Use the factors to write the factored form of the expression:Next, look for two factors of $c$, or 48, that have a sum of $b$, or –14. When $c$, or 48, is positive, both factors will have the same sign as $b$. So, the factors are –6 and –8.

Use the factors to write the factored form of the expression:Next, look for two factors of $c$, or –2 that have a sum of $b$, or 1. When $c$ is negative, the factors will have different signs. The factor with the greater absolute value will have the same sign as $b$. The factors are $+2$ and $-1$.

Use the factors to write the factored form of the expression:Next, look for two factors of $b$, or –21, that have a sum of $c$, or –4. When $c$ is negative, the factors will have different signs. The factor with the greater absolute value will have the same sign as $b$. The factors are $-7$ and $+3$.

Use the factors to write the factored form of the expression:### Factoring with a Leading Coefficient

To factor the expression, factor out –1 from each term if necessary so that $a$ is positive. Then look for two factors of $ac$ that have a sum of $b$. When $ac$ is positive, the two factors have the same sign as determined by the sign of $b$. When $ac$ is negative, the two factors have opposite signs, and the factor with the greater absolute value, or the distance from zero on a number line, will have the same sign as $b$. Use the two factors to write the middle term of the quadratic expression as a sum, and then factor by grouping.

To solve a quadratic equation by factoring by grouping:

1. Set one side of the equation equal to zero.

2. Factor the other side of the equation by grouping.

3. Set each factor equal to zero.

4. Solve the resulting equations.

### Special Products

### Special Products of Quadratic Expressions

Perfect Square Trinomial | Difference of Squares |
---|---|

$a^2+2ab+b^2=(a+b)(a+b)=(a+b)^2$ $a^2-2ab+b^2=(a-b)(a-b)=(a-b)^2$ |
$a^2-b^2=(a+b)(a-b)$ |

### Substitution Methods

### Factoring Methods

Method | When to Use |
---|---|

Greatest Common Factor (GCF) | Use with any number of terms. Look for a greatest common factor of the terms, and factor out the GCF if possible.
$ac+bc=a(b+c)$ |

Difference of squares | Use when there are two terms that match the pattern for a difference of squares.
$a^2-b^2=(a+b)(a-b)$ |

Perfect square trinomial | Use when there are three terms that match the pattern for a perfect square trinomial.
$\begin{aligned}a^2+2ab+b^2&=(a+b)(a+b)=(a+b)^2\\a^2-2ab+b^2&=(a-b)(a-b)=(a-b)^2\end{aligned}$ |

Trinomial factoring with $a=1$ | Use when there are three terms in the form $x^2+bx+c$. Look for factors of $c$ that have a sum of $b$. Use those values to write factors of the expression.
$x^2+bx+c=(x+\square)(x+\square)$ |

Trinomial factoring with $a\neq1$ | Use when there are three terms in the form $ax^2+bx+c$ with $a\neq1$. Look for factors of the product of $a$ and $c$ that have a sum of $b$. Use those values to write the middle term as a sum. Then factor by grouping.
$ax^2+bx+c=ax^2+\square x+\square x+c=(\square x+\square)(\square x + \square)$ |

Grouping | Use when a quadratic trinomial has been written with four terms, such as when the middle term is written as a sum.
$\begin{aligned}ax+bx+ay+by&=x(a+b)+y(a+b)\\&=(a+b)(x+y)\end{aligned}$ |

Substitution | Use when a part of the expression can be replaced by a temporary variable so that another factoring method applies.
$\begin{aligned}x^4+5x^2-36&=u^2+5u-36\\&=(u-4)(u+9)\\&=(x^2-4)(x^2+9)\\&=(x+4)(x-4)(x^2+9)\end{aligned}$ |