Factoring to Find the Zeros

Factoring Trinomials Into Binomials

Day 1

Vocabulary

• a value - the coefficient of the x² term

• b value - the coefficient of the x term

• c value - just the constant; the number by itself

• "zeros" - These can also be called the "solutions" to a quadratic function. They are called zeros because it refers to the x-values that give you a y-value of 0

We can use factoring to help us find the zeros (aka solutions) of quadratics that have a=1

Step 1: Check that the a-value is 1

Step 2: List out the pairs of factors for the c-value. Don't forget to include positive and negative values.

Step 3: Find the factors that add to the b-value

Step 4: Write the binomials

If you want to solve:

Step 5: Set each one equal to 0 and solve for x

Practice! Practice! Practice

• Step 1: Check that the a-value is 1

• Step 2: List out the factors for the c-value

• Step 3: Find the factors that add to the b-value

• Step 4: Write the binomials

• If you want to solve:

  Step 5: Set each one equal to 0 and solve for x

Factoring

​Day 2: By GCD

Today's Vocabulary

• Coefficient - a number in front of a variable

• Variable - a letter standing in for an unknown value

• Term - composed of a number and variable; terms are separated by addition or subtraction

• Monomial - one term

• Binomial - Two terms

• Polynomial - Multiple terms

Today's Vocabulary cont.

• Factor - a number than can divide a given number

• Greatest common factor - the biggest number or term that can divide EVERY number or term in the expression

Examples

1. ​Write the prime factorization of each term. This includes the variables

2. check for any numbers and variables that ALL terms have in common

3. Bring out the common terms and keep the rest in parentheses.

NOTE: Sometimes you will need to factor out the GCF before factoring the quadratic.

Examples

1. ​Write the prime factorization of each term. This includes the variables

2. check for any numbers and variables that ALL terms have in common

3. Bring out the common terms and keep the rest in parentheses.

NOTE: Sometimes you will need to factor out the GCF before factoring the quadratic.

Quadratic Formula

​Day 3

Main Ideas This Week

• Factoring GCF comes from every term in your expression

• Quadratic expressions have a squared variable

• Factoring a trinomial into the product of two binomials let's us find "zeros" aka solutions to the expression.

...but what do we do when the a-value is not 1 or if I can't factor the expression?

NEW Strategy today!

The Quadratic Formula

The a,b and c values will come from the expression you are given

• Step 1: Identify the a, b, and c values

• Step 2: Plug them into their corresponding places in the formula

• Step 3: Follow PEMDAS to simplify

• Step 4: The + and - symbols create two expressions

Let's Mix-It Up!!

Review Day and Practice Problems

Reviewing GCF

• Greatest common factor - the biggest number or term that can divide EVERY number or term in the expression

Multiple Choice

Factor the expression using the GCF $$8x+16$$  

• $$2\left(4x+8\right)$$  

• $$4\left(2x+4\right)$$  

• $$8$$  

• $$8\left(x+2\right)$$  

Multiple Choice

Complete the Factoring  (Hint: GCF) $$5x^2-40x\ =\ 5x\left(?\right)$$  

• $$x-8$$  

• $$8-x$$  

• $$x^2-8$$  

• $$8-x^2$$  

Multiple Choice

Factor using the GCF

$$3x^2y^5+6x^2y^4$$  

• $$3x^4y^2\left(y+2\right)$$  

• $$3x^2y^4\left(y+2\right)$$  

• $$x^2y^4\left(3y+2\right)$$  

• $$y+2$$  

Factoring quadratics into the product of two binomials

Greatest common factor - the biggest number or term that can divide EVERY number or term in the expression

Step 1: Check that the a-value is 1

Step 2: List out the factors for the c-value

Step 3: Find the factors that add to the b-value

Step 4: Write the binomials

Step 5: Set each one equal to 0 and solve for x

Multiple Choice

Factor Completely

$$s^2+5s+6$$  

• $$\left(s-3\right)\left(s-2\right)$$  

• $$\left(s+3\right)\left(s+2\right)$$  

• $$\left(s+6\right)\left(s-1\right)$$  

• $$\left(s+6\right)\left(s+5\right)$$  

Multiple Choice

Factor Completely

$$x^2+3x-54$$  

• $$\left(x-54\right)\left(x+1\right)$$  

• $$\left(x-6\right)\left(x+9\right)$$  

• $$\left(x+6\right)\left(x-9\right)$$  

• $$\left(x+54\right)\left(x-1\right)$$  

Multiple Choice

Factor Completely

$$9+6x+x^2$$  

• $$\left(x+3\right)\left(x+3\right)$$  

• $$\left(x-3\right)\left(x+3\right)$$  

• $$\left(x-3\right)\left(x-3\right)$$  

• $$\left(x+1\right)\left(x+9\right)$$  

The Quadratic Formula

The a,b and c values will come from the expression you are given. a is always next to the term with the exponent of 2, b is the one matching with the variable and c is the number by itself

Multiple Choice

Find the solutions to the following quadratic expression

(Hint: Use quadratic Formula)

$$-2m^2+m+6$$  

• $$x=-2,\ x=3$$  

• $$x=-6,\ x=2$$  

• $$x=2,\ x=-1.5$$  

• $$x=-3,\ x=4$$  

Multiple Choice

Find the solutions to the following quadratic expression

(Hint: Use quadratic Formula)

$$3a^2-6a+3$$  

• $$x=-2,\ x=3$$  

• $$x=2$$  

• $$x=-3$$  

• $$x=1$$