Note-Taking Guides. How to use these documents for success
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1 1 Note-Taking Guides How to use these documents for success Print all the pages for the module. Open the first lesson on the computer. Fill in the guide as you read. Do the practice problems on notebook paper (usually). Put the notes and practice problems in a notebook. You can use these anytime! Review the notes before you go to sleep. Short term memory is converted to long term memory ONLY while you sleep. Your brain starts at the end of your day and converts things to long term memory in reverse order.
2 2 Module # GCF Factoring Fill in the factoring flowchart below: Factoring out the GCF There are three steps to finding the GCF and factoring an expression. They are:
3 3 Example 1: 2x + 6 Step 1 Identify GCF Step 2 Factor out GCF Step 3 Write the answer Example 2: 10x x 2 Step 1 Identify GCF Step 2 Factor out GCF Step 3 Write the answer Example 3: 8x 3-4x 2 10x Step 1 Identify GCF Step 2 Factor out GCF Step 3 Write the answer
4 4 Example 4: -6x 4 y 9x 3 y 2 + 3x 2 y 3 If the first term in the expression you are factoring is negative, be sure to: Step 1 Identify GCF Step 2 Factor out GCF Step 3 Write the answer Special Note A prime number has only two factors. They are: If an expression does not have a common factor, it is called a.
5 Difference of Squares A difference of squares is a polynomial that: Perfect Squares Perfect squares are numbers that: The exponents on variables of perfect squares are: The formula for factoring the difference of squares is: Remember: if you have a GCF in a difference of squares problem you must factor that out first. It s always step 1 in our factoring process. Example 1: Factor 9x 2 4 Step 1 Identify the perfect squares Step 2 Use formula to factor Example 2: Factor x 2 36y 4 Step 1 Identify the perfect squares Step 2 Use formula to factor
6 6 Example 3: Factor 2x 2 y 2 50 Step 1 Identify the perfect squares Step 2 Use formula to factor If you have a GCF in a problem, you must
7 Factoring by Grouping The four steps to factoring by grouping are: Example 1: Factor 7x 3-28x 2 + 3x - 12 Step 1 Divide the polynomial Step 2 Factor out GCF (first terms) Step 3 Factor out GCF (last terms) Step 4 Factor the parentheses Example 2: Factor 6x 3 + 3x 2-2x - 1 Step 1 Divide the polynomial Step 2 Factor out GCF (first terms) Step 3 Factor out GCF (last terms) Step 4 Factor the parentheses
8 8 Example 3: Factor 60ab 20bx 30ax + 10x 2 Step 1 Divide the polynomial Step 2 Factor out GCF (first terms) Step 3 Factor out GCF (last terms) Step 4 Factor the parentheses Parentheses don t match! The two possible reasons for the parenthesis not to match in this process are: 1. 2.
9 Factoring Trinomials Part 1 Factoring Trinomials of the form x 2 + bx + c Factoring trinomials is like what process in reverse? to Factoring Trinomials There are four steps to factoring trinomials of the form x 2 + bx + c: Example 1: Factor x 2 + 6x + 8 Step 1 Identify all the factor pairs for the last term Step 2 Select the pair of factors that add up to the middle term Step 3 Determine the signs Step 4 Check your guess using FOIL
10 10 Example 2: Factor x 2-12x + 35 Step 1 Identify all the factor pairs for the last term Step 2 Select the pair of factors that add up to the middle term Step 3 Determine the signs Step 4 Check your guess using FOIL Example 3: Factor x 2-3x - 18 Step 1 Identify all the factor pairs for the last term Step 2 Select the pair of factors that add up to the middle term Step 3 Determine the signs Step 4 Check your guess using FOIL
11 11 Example 4: Factor x 2 + 2xy + y 2 Step 1 Identify all the factor pairs for the last term Step 2 Select the pair of factors that add up to the middle term Step 3 Determine the signs Step 4 Check your guess using FOIL Sign Rules for Factoring Trinomials If the last term in the trinomial is positive: If the last term of the trinomial is negative: If the middle term is positive: If the middle term is negative: If the last term in the polynomial is negative:
12 Factoring Trinomials Part 2 Factoring Trinomials of the form ax 2 + bx + c What is a leading coefficient? Factoring by Guessing and Checking There are four steps to factoring trinomials of the form ax 2 + bx + c by guessing and checking: Example 1: Factor 8x 2 + x - 7 Step 1 Determine signs Step 2 Determine factors of first term Step 3 Determine factors of last term Step 4 Check your factors using FOIL
13 13 Factoring by Grouping There are six steps to factoring trinomials by grouping: Example 2: Factor 2x 2-7x + 6 Step 1 Multiply first and last coefficients Step 2 List all factor pairs Step 3 Determine factors that multiply to give last coefficient but add to give middle coefficient Step 4 Write original trinomial as 4-term polynomial Step 5 Factor by grouping Step 6 Check using FOIL
14 14 Example 3: Factor 5x 2 + 8xy - 4y 2 Step 1 Multiply first and last coefficients Step 2 List all factor pairs Step 3 Determine factors that multiply to give last coefficient but add to give middle coefficient Step 4 Write original trinomial as 4-term polynomial Step 5 Factor by grouping Step 6 Check using FOIL Example 4: Factor 8x 2 14x + 5 Step 1 Multiply first and last coefficients Step 2 List all factor pairs Step 3 Determine factors that multiply to give last coefficient but add to give middle coefficient Step 4 Write original trinomial as 4-term polynomial Step 5 Factor by grouping Step 6 Check using FOIL
15 Factoring Completely Example 1: 10x 3 40x Step 1 Step 2 Step 3 Example 2: y 5-2y 4-35y 3 Step 1 Step 2 Step 3
16 16 Example 3: 2x 2-11x + 12 Step 1 Step 2 Step 3 Example 4: 3m 4-3 Step 1 Step 2 Step 3 Example 5: 2x 4 5x 2-12 Step 1 Step 2 Step 3
17 Solving Equations by Factoring Review of Linear Equations Linear equations look like:. The highest power in a linear equation is:. Linear equations have and on the variables. The graph of a linear equation is:. Quadratic Equations A quadratic equation looks like: The highest power in a quadratic equation is: Quadratic equations have and as the highest power of x. The solutions to quadratic equations are the points where: The graph of a quadratic equation is: Solving Quadratic Equations Quadratic equations can be written in two forms: In the second, the y is replaced with:. Quadratic equations can have,, or no solutions. Most of the time, there are solutions. The solutions are the of the parabola. Solving Quadratic Equations by Factoring There are three General to solving quadratic equations by factoring. Step 1. Step 2. Step 3.
18 18 Explain the property for zero that explains why we can solve quadratic equations by factoring: Example 1: 3x 2 = 12x Step 1 Set equation equal to zero Step 2 - Factor Step 3 Set factors equal to zero Example 2: x 2 7x + 12 = 0 Step 1 Set equation equal to zero Step 2 - Factor Step 3 Set factors equal to zero Example 3: 6x 2 9x = 60 Step 1 Set equation equal to zero Step 2 - Factor Step 3 Set factors equal to zero
19 19 Example 4: x 2 + 6x + 9 = 60 Step 1 Set equation equal to zero Step 2 - Factor Step 3 Set factors equal to zero
20 Quadratic Formula The quadratic formula allows you to find the values of x even when the quadratic polynomial is not factorable. The Quadratic Formula The quadratic formula looks like: The equation must be set equal to: The equation should be placed in: The values for a, b and c are pulled from the quadratic equation being solved. 1. a is 2. b is 3. c is Follow the order of operations when simplifying the quadratic formula. When do most students make mistakes with the quadratic formula? Example 1: 2x 2 + 3x 6 = 0 Step 1 Identify a, b and c Step 2 Substitute a, b and c in the formula Step 3 Write the two solutions for x
21 21 Example 2: x 2 4x - 3 = 0 Step 1 Identify a, b and c Step 2 Substitute a, b and c in the formula Step 3 Write the two solutions for x Example 3: 2x 2 + 5x = 3 Step 1 Identify a, b and c Step 2 Substitute a, b and c in the formula Step 3 Write the two solutions for x Example 4: x 2-2x + 1 = 0 Step 1 Identify a, b and c Step 2 Substitute a, b and c in the formula Step 3 Write the two solutions for x
22 Graphing Quadratic Functions Sketching Graphs of Quadratics by Hand There are 4 steps to graphing a quadratic function by hand: To calculate the x-coordinate of the vertex, use the formula:. Example 1 Sketch the graph of the quadratic function y = 5x 2 Step 1 Standard Form Step 2 Find vertex Step 3 Table of Values x y
23 23 Step 4 Plot and sketch Example 2 Sketch the graph of the quadratic function y = x 2 + 4x + 3 Step 1 Standard Form Step 2 Find vertex Step 3 Table of Values x y
24 24 Step 4 Plot and sketch Example 3 Sketch the graph of the quadratic function y = -4x 2 + 8x - 1 Step 1 Standard Form Step 2 Find vertex Step 3 Table of Values x y Step 4 Plot and sketch
25 25 Graphing Quadratics using Technology Go to the website and download the software. Why do we use technology to graph quadratic equations? The solution to a quadratic function is/are:
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