Solving One-Step Equations



Solving One-Step Equations |Solving Multi-Step Equations | |

|Multi-Step Equations w/ Distributive Property |Multi-step Equations with Variables on Each Side |

|Percent of Change |Solving Equations with Formulas |

|Relations | |

|Midpoint of a Line |Linear Equations |

|How To Graph a Line |Conversions Between Standard Form and Slope-Intercept Form |

|Finding the Slope of a Line | |

|Parallel and Perpendicular Lines |Line of Best Fit |

|Polynomial Operations |Power to a Power |

|Multiplying a Polynomial by a Monomial |binomials times binomials |

end

2/8/2012

Solving One-Step Equations

|4 + ( = 10 |Algebra |

|( ( ( ( ( | |

|( ( ( ( ( ( |Concrete example |

|( ( ( | |

|4 + x = 10 |Algebra with a variable |

|To solve: | 4 + x = 10 |(“=” means “same as”) |

| |-4 - 4 | |

| | 0 + x = 6 |This means that 6 is the only number that can replace x to make the equation true. |

| |x = 6 | |

ALWAYS SHOW YOUR WORK!!

|NO! |YES! |ASK YOURSELF: What is happening to the variable? How can I get the |

|6 = X + 2 |6 = X + 2 |variable by itself? |

|4 = X |-2 -2 | |

| | 4 = X | |

Examples: Addition/Subtraction

Notice: In these examples, you are moving an entire quantity (the constant in these examples) to the other side of the equation.

|1) | 27 + n = 46 |2) |-5 + a = 21 |

| |-27 -27 |Remember that whatever you |+5 + 5 |

| | 0 + n = 19 |do on one side of the equation you have to do on | |

| |n = 19 |the other! |a = 26 |

Examples: Multiplication/Division

Notice: In these examples, you are separating the quantity (the coefficient and the variable).

|3) |- 8n = -64 |4) | m = 12 |

| | |Remember that whatever you do on one side of the |7 |

| |-8n = -64 |equation you have to do on the other! | |

| |-8 -8 | |7 * m = 12(7) |

| | | |1 7 |

| |n = 8 | | |

| | | |m = 84 |

|5) | 5b = 145 |6) | -13 = m |

| | | |-5 |

| |5b = 145 | | |

| |5 5 | |(-13)(-5) = m * -5 |

| | | |-5 |

| |b = 29 | |65 = m |

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2/9/2012

Solving Multi-Step Equations

Compare Equations

|Equation |Comments |Equation 2 |

| X + 3 = 15 |( Total is the same ( | 2x + 3 = 15 |

| |Variable has a coefficient greater than 1 ( | |

|X + 3 = 15 | |2x + 3 = 15 |

|-3 -3 | |-3 = -3 |

| X = 12 |( One step | 2x = 12 |

| | | |

| | |2x = 12 |

| | |2 |

| | | |

| |Two steps ( |X = 6 |

Coefficient = the number in front of a variable

Constant = a number. Example: in 2x + 3, 3 is the constant.

Like Terms = quantities with the same variable (eg 2x, 0.5x, 644x, 342909835x, 1/2x, etc, are like terms because they all have an x). Numbers without a variable (constants) are also like terms!

How to Solve Multi-Step Equations:

1. Combine like terms on the left or right if you can.

2. Get all your variables on one side and all your constants on the other side (addition/subtraction OR multiplication/division)

3. Divide both sides by the coefficient of the variable.

Examples:

|A) | 7 – y – y = -1 |

|1. Combine like terms. |7 – 2y = -1 |

|2. Get all variables on one side and all constants on the other. | 7 – 2y = -1 |

| |-7 -7 |

| |-2y = -8 |

|3. Divide both sides by the coefficient of the variable. |-2y = -8 |

| |-2 -2 |

| | |

| |y = 4 |

|B) | | 13 = 5 + 3b - 13 |

|1. Combine like terms. | |13 = 3b - 8 |

|2. Get all variables on one side and all constants on the | | 13 = 3b – 8 |

|other. | |+8 +8 |

| | |21 = 3b |

|3. Divide both sides by the coefficient of the variable. | |21 = 3b |

| | |3 3 |

| | | |

| | |7 = b |

|C) | a – 18 = 2 |z + 10 = 2 |

| |5 |9 |

|1. Combine like terms. | | |

|2. Get all variables on one side and all constants on the | a – 18 = 2 |9(z + 10) = 2(9) |

|other. |5 |9 |

| |+18 +18 | |

| |a = 20 |z + 10 = 18 |

| |5 | |

| | | z + 10 = 18 |

| | |-10 -10 |

| | |z = 8 |

|3. Divide both sides by the coefficient of the variable. | a * 5 = 20 * 5 | |

| |5 | |

| | | |

| |a = 120 | |

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2/10/2012

Solving Multi-Step Equations That Include the Distributive Property

| |Example: 2 = -2(n – 4) |

|Step 1: Distribute First |2 = -2(n) + (-2)(-4) |

| |2 = -2n + 8 |

|Step 2: Solve as you would any other multi-step equation | - 8 = -2n – 8 |

| |- 6 = -2n |

| | |

| |-6 = -2n |

| |-2 -2 |

| | |

| |3 = n |

|-5(x + 3) = -45 | |

|-5x + (-15) = -45 | |

|+15 +15 |This means that 6 is the only number that you can substitute to make this equation true. |

|-5x = -30 |( |

|x = 6 | |

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2/14/2012

Solving Multi-Step Equations with Variables on Each Side

|Step 1: Add like terms (there are none) |Example: 6 – x = 5x + 30 |

|Step 2: Distribute (none to do) |- 5x -5x |

|Step 3: Variables on the left |6 – 6x = 30 |

|Step 4: Numbers (constants) on the right | -6 -6 |

| |-6x = 24 |

| |-6 -6 |

| | |

| |x = 4 |

|Step 1: Add like terms (there are none) |Example: 5x + 2 = 2x - 10 |

|Step 2: Distribute (none to do) |- 2x -2x |

|Step 3: Variables on the left |3x + 2 = - 10 |

|Step 4: Numbers (constants) on the right | -2 -2 |

| |3x = -12 |

| |3 3 |

| | |

| |x = 4 |

| |Example: 5y – 2y = 3y + 2 |

|Step 1: Add like terms |3y = 3y + 2 |

|Step 2: Distribute (none to do) |- 3y -3y |

|Step 3: Variables on the left (if possible) |0 = 2 |

|Step 4: Numbers (constants) on the right |0 does not equal 2. It is a false statement. This means that there are no |

| |real numbers that can be substituted for y in this equation. |

|Step 1: Add like terms (there are none) |Example: 2y + 4 = 2(y + 2) |

|Step 2: Distribute |2y + 4 = 2(y) + 2(2) |

| |2y + 4 = 2y + 4 ( |

|Step 3: Variables on the left |-2y -2y |

| |0 + 4 = 0 + 4 ( |

|Step 4: Numbers (constants) on the right |Both sides of the equation are equal without a variable. Therefore, any real |

| |number can be substituted for y, and the equation will be true. |

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2/16/12

Percent of Change

|Types of INCREASES |Types of DECREASES |

|New = original(1 + r) |New = original(1-r) |

|* Tax (for example, on clothing purchases) |* sales |

|* Interest (such as on a bank account) |* discounts |

|* inflation |* depreciation |

|* commission (such as when a car salesman earns a certain % of what he sells) | |

When solving a Percent of Change problem:

| | |EXAMPLE: original: 18 new: 10 |

|Step 1: |WRITE OUT THE FORMULA!!!! | New = original(1-r) | |

| |If you don’t it will be counted as wrong! | | |

|Step 2: |Organize your information. | |N =10 |

| | | |O = 18 |

| | | |r = ? |

|Step 3: |Plug your values into the equation. | 10 = 18(1-r) | |

|Step 4: |Solve the equation. | 10 = 18 – 18r | |

| | |-18 = -18 | |

| | |- 8 = - 18r | |

| | |-18 -18 | |

| | | | |

| | |.4444 = r | |

| | |44% = r | |

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2/21/2012

Solving Equations with a Formula

| | |EXAMPLE: P = 2L + 2W |

| | |If the length is 5 and the perimeter is 24, what is the value of W?|

|Step 1: |WRITE OUT THE FORMULA!!!! |P = 2L + 2W | |

|Step 2: |Organize your information. | |P = 24 |

| | | |L = 5 |

| | | |W = ? |

|Step 3: |Plug your values into the equation. | 24 = 2(5) + 2W | |

|Step 4: |Solve the equation. | 24 = 10 + 2W | |

| | |-10 =-10 + 2W | |

| | |14 = 2W | |

| | |2 2 | |

| | | | |

| | |7 = W | |

| | |EXAMPLE: P = 2L + 2W |

| | |If the width is 2.6 and the perimeter is 12.4, find L. |

|Step 1: |WRITE OUT THE FORMULA!!!! |P = 2L + 2W | |

|Step 2: |Organize your information. | |P = 12.4 |

| | | |L = ? |

| | | |W = 2.6 |

|Step 3: |Plug your values into the equation. | 12.4 = 2L + 2(2.6) | |

|Step 4: |Solve the equation. | 12.4 = 2L + 5.2 | |

| | |- 5.2 -5.2 | |

| | |7.2 = 2L | |

| | |2 2 | |

| | | | |

| | |3.6 = L | |

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RELATIONS

Domain = x values

Range = y values

f(x) = 2x2 + 7x – 2

f(2a) = 2(2a)2 + 7(2a) – 2

= 2(4a)2 + 14a – 2

= 8a2 + 14a – 2

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Midpoint of a Line

The Midpoint of a Line is the middle, center, or halfway point of a line. Both segments of the line are equal distances.

Midpoint = m x + x , y + y

2. 2

Example: Find the midpoint between (2, -4) and (-10, -3)

1. Identify your x and y coordinates: x y x y

(2, -4) (-10, -3)

2. Put into the formula: 2 + (-10) , -4 + (-3)

2 2

-8 , -7

2 2

(-4, -7/2) or (-4, -3.5) or (-4, -3½)

Example: Using the midpoint formula, we will find the missing information:

Given endpoint (-3, -7)

Midpoint (-8, 4)

Find endpoint (x, y)

Solving by formula:

x + x = xm , y + y = ym

2 2

2(-3 + x) = -8(2) , 2(-7 + y) = 4(2)

2 2

-3 + x = -16 , -7 + y = 8

+3 +3 +7 +7

x = - 13 , y = 15

(-13, 15)

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3/12/2012

Linear Equations

* What makes an equation a linear equation? Linear = makes a line

* What does linear look like?

1. No multiplication between variables together (in other words, no 2 variables are multiplied together)

2. No exponents greater than 1

| |Equation |Linear or Non-Linear |Explanation |

|A. |2x = 3y + 1 |Linear |No multiplication between variables; no exponents |

| | | |> 1 |

|B. |4xy + 2y = 7 |Non-linear |xy |

|C. |2x2 = 4y - 3 |Non-linear |x2 |

|D. |X – 4y = 2 |Linear |No multiplication between variables; no exponents |

| |5 5 | |>1 |

Linear equations can be written in standard form (Ax + By = C).

X-intercept –A point (x, y) where a line crosses the x-axis (x, 0)

Y-intercept – A point (x, y) where a line crosses the y-axis (0, y)

How do you graph a line?

1. Standard Form (Ax + By = C): use x and y intercepts

2. Slope-Intercept Form (y = mx + b): use graphing calculator:

y =

2nd graph

Plot

1. Graphing a line using Standard Form

Plot the x-intercept

Plot the y-intercept

| |Example: Graph 2x + 5y = 20 | |

| | | |

|Write your equation twice. |2x + 5y = 20 |2x + 5y = 20 |

|Cover (mark out) the x and its coefficient in one; | | |

|cover (mark out) the y and its coefficient in the |5y = 20 |2x = 20 |

|other |5 |2 |

|Solve. | | |

|Write your ordered pairs. |y = 4 |x = 10 |

|Plot your points on a graph and draw your line |x-intercept: (0, 4) |y-intercept: (10, 0) |

|through the points. | | |

| |Example: Graph 3x + y = -1 | |

| | | |

|Write your equation twice. |3x + y = -1 |3x + y = -1 |

|Cover (mark out) the x and its coefficient in one; | | |

|cover (mark out) the y and its coefficient in the |y = -1 |3x = -1 |

|other | |3 |

|Solve. |x-intercept: (0, -1) | |

|Write your ordered pairs. | |x = -1/3 |

|Plot your points on a graph and draw your line | |y-intercept: (-1/3, 0) |

|through the points. | | |

| |Example: Graph x – y = -3 | |

| | | |

|Write your equation twice. |x - y = -3 |x - y = -3 |

|Cover (mark out) the x and its coefficient in one; | | |

|cover (mark out) the y and its coefficient in the |-y = -3 |x = -3 |

|other |You can’t have a –y, so multiply both sides by -1: |y-intercept: (-3, 0) |

|Solve. |-y(-1) = (-3)(-1) | |

|Write your ordered pairs. |y = 3 | |

|Plot your points on a graph and draw your line |x-intercept: (0, 3) | |

|through the points. | | |

2. Graphing a line using Slope-Intercept Form

y = mx + b

On your calculator, press:

Y =

2nd

graph (table)

Pick out 3 or 4 points from the table that are easy to plot (ie X and Y are whole numbers) and plot them on a graph. Draw your line through those points.

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3/14/12

Conversions Between Standard Form and Slope-Intercept Form

1. Converting from Standard Form to Slope-Intercept Form

Ax + By = C ( y = mx + b

Example: Convert 4x + y = -2 to Slope-Intercept Form

|Write your original formula. | 4x + y = -2 – 4x |NOTE: You are not separating the coefficient from the variable (the x |

|Get y by itself. |-4x |from the 4), so you are not dividing both sides by 4. You are moving the|

| |0 |entire quantity of 4x to the other side of the equation, which is why you|

| |y = -4x -2 |subtract. In other words, you are moving the variable, not isolating it. |

| | |NOTE: Write the sentence so that it follows the form. Therefore, you |

| | |write y = -4x – 2, rather than y = -2 – 4x. |

Example: Convert 5x – 3y = -6 to Slope-Intercept Form

|Write your original formula. | 5x – 3y = -6 – 5x |NOTE: You are not separating the coefficient from the variable (the x |

|Get y by itself (move x). |-5x |from the 5), so you are not dividing both sides by 5. You are moving the|

| |0 |entire quantity of 5x, which is why you subtract. In other words, you are|

| |-3y = -5x -6 |moving the variable, not isolating it. |

|3. Solve so that y is positive and has a coefficient|-3 -3 -3 |NOTE: Write the sentence so that it follows the form. Therefore, you |

|of 1 (isolate y and make it positive). | |write -3y = -5x – 6, rather than -3y = -6 – 5x. |

| |y = 5/3x + 2 |NOTE: In this step you ARE separating the coefficient from the variable |

| | |(the -3 from the y), so you DO divide. In other words, you are isolating |

| | |the variable, not moving it. |

| | |NOTE: Write the slope (m) as a FRACTION, not a decimal. |

Example: Convert 10x – y = 6 to Slope-Intercept Form

|Write your original formula. | 10x – y = 6 – 10x |NOTE: You are not separating the coefficient from the variable (the x |

|Get y by itself (move x). |-10x |from the 5), so you are not dividing both sides by 10. You are moving |

| |0 |the entire quantity of 10x, which is why you subtract. In other words, |

| |(-1) -y = (-1)-10x + (-1)6 |you are moving the variable, not isolating it. |

|3. Solve so that y is positive and has a coefficient| |NOTE: Write the sentence so that it follows the form. Therefore, you |

|of 1 (isolate y and make it positive). |y = 10x + 6 |write -y = -10x – 6, rather than -y = -6 – 10x. |

2 Converting from Slope-Intercept Form to Standard Form

y = mx + b ( Ax + By = C

To be standard form:

1. No fractions anywhere (includes decimals): all whole numbers

2. Leading x cannot be negative

3. Must be in simplest form

4x + 2y = 10 ( 2x + y = 5

2 2 2

Example 1: Change y = 3/2x – 5 to Standard Form (Ax + By = C)

– 3/2x + y = (3/2x) – (3/2x) – 5

– 3/2x + y = -5 ( This is negative and a fraction, so it is not yet in standard form.

(-2)(-3/2)x + (-2)y = -5(-2)

-3x -2y = 10

Example 2: Change y = 1/5x – 1 to Standard Form (Ax + By = C)

– 1/5x + y = (1/5x) – (1/5x) – 1

– 1/5x + y = -1 ( This is negative and a fraction, so it is not yet in standard form.

(-5)(-1/5)x + (-5)y = -1(-5)

x -5y = 10

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Finding the Slope of a Line

Slope is rate of change

• It is the steepness of a line

• Represented by a fraction

m = slope = rise = ( y = change in y

run ( x change in x

m = 4 = 4

1

Positive slope

Negative slope

Two ways to find slope:

1. graph

2. formula m = y – y

x – x

Example: Find the slope of the line going through the points (-4, 4) and (4, 6)

| m = y – y | m = y – y |

|x – x |x – x |

| | |

|4 – 6 = -2 = 1 |6 - 4 = 2 = 1 |

|-4 – 4 -8 4 |4-(-4) 8 4 |

Example: Find the slope of the line going through the points (-4, -1) and (-2, -5)

m = y – y = -1 – (-5) = 4 = -2

x – x -4 – (-2) -2

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3/16/12

PARALLEL AND PERPENDICULAR LINES

Parallel Lines (||)

• Do not intersect

• Have the same slopes

Example: Write the slope-intercept form of an equation of a line parallel to y = 4x – 2 and passing through the point (-2, 2)

|1. Use the formula |y – y1 = m(x – x1) | |

|y – y1 = m(x – x1) | | |

|2. Organize your information | |m = 4 x1 = -2 y1 = 2 |

|3. Plug your values into your formula. |y – 2 = 4(x – (-2)) | |

|4. Solve |y – 2 = 4(x + 2) | |

| |y – 2 = 4x + 8 | |

| |+2 +2 | |

| |y = 4x + 10 | |

Perpendicular Lines (()

• Intersect

• Slopes are 1) opposite (change the sign)

2) reciprocal (flip them)

Example: Write the slope-intercept form of the ( line y = 1/2x + 1, crossing through point (4, 2).

|1. Use the formula |y – y1 = m(x – x1) | |

|y – y1 = m(x – x1) | | |

|2. Convert your slope | | Opp flip |

| | |m = ½ ( - ½ ( - 2/1 = -2 |

|3. Organize your information | |m = -2 x1 = 4 y1 = 2 |

|4. Plug your values into your formula. |y – 2 = -2(x – 4) | |

|5. Solve |y – 2 = -2x + 8 | |

| |+2 +2 | |

| |y = -2x + 10 | |

Example: Write the slope-intercept form of the ( line 2x + 4y = 12, crossing through point (-1, 3).

|1. Change the form of the original line from standard form | 2x + 4y = 12 - 2x | |

|to slope-intercept form. |-2x | |

| |4y = -2x – 12 | |

| |4 4 | |

| | | |

| |y = -1/2x - 3 | |

|1. Use the formula |y – y1 = m(x – x1) | |

|y – y1 = m(x – x1) | | |

|2. Convert your slope | | Opp flip |

| | |m = -½ ( ½ ( 2/1 = 2 |

|3. Organize your information | |m = 2 x1 = -1 y1 = 3 |

|4. Plug your values into your formula. |y – 3 = 2(x – (-1)) | |

|5. Solve |y – 3 = 2x + 2 | |

| |+3 +3 | |

| |y = 2x + 5 | |

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3/19/12

Line of Best Fit

Inside a SCATTERPLOT are data (ordered pairs). The more data you have, the better your line will be.

Why do we need a line? TO PREDICT! (for example, it helps business owners predict profits, whether to expand or downsize, etc).

|Line of Best Fit |( |Equation of a line to predict. |

|Regression Equation | | |

|Prediction Equation | | |

|Best Fit Line | | |

|X (L1) |Y (L2) |

|Independent Variable |Dependent Variable |

** See “Best-Fit Line” Handout**

Types of Questions you can be asked with Line of Best Fit

1. To find the equation

2. To make a prediction (from a line)

3. What does slope mean/represent?

4. What does y-intercept represent?

Example: The cab driver charges a $5 flat fee and $.25 per mile.

y = .25x + 5

y represents the total bill

.25 represents the charge or RATE

x represents per mile

5 represents the flat fee or starting point; the total bill starting at 0 miles

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4/9/12

Polynomial Operations

Monomial: one term

A number, a variable, or the product of 1 or more variables

Examples: 5, a, 6x, 5x2yz

Binomial: two terms

Addition or subtraction of two monomials

Examples: 5-5x; 2x2+3

Trinomial: three terms

Addition or subtraction of three monomials

Example: 6+7x2-3x

Adding Polynomials:

1) (4a - 5) + (3a + 6)

Combine like terms/regroup

(4a - 5) + (3a + 6)

4a + 3a - 5 + 6

7a + 1

2) (6xy + 2y + 6x) + (4xy – x)

10xy + 2y + 5x

Subtracting Polynomials:

|1) |(3a – 5) – (5a + 1) |

|DISTRIBUTE the -1 on the 2nd monomial |(3a – 5) + (- 5a – 1) |

|Combine like terms |- 2a - 6 |

|2) |(9xy + y – 2x) – (6xy – 2x) |

|DISTRIBUTE the -1 on the 2nd monomial |(9xy + y – 2x) +(- 6xy + 2x) |

|Combine like terms |3xy + y + 0 |

| |3xy + y |

| | |

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4/10/12

Multiplying Monomials

1. Multiply coefficient (whole number)

2. Add exponents

What does y3 mean? y * y * y

What does (x2)(x4) mean? x * x |* x * x * x * x ( x6

x2+4 = x6

EXAMPLES:

1) (x16) (x163) = x16+163 = x179

2) (4x3y2) (-2xy)

(4) (-2) x3+1 y2+1

-8 x4 y3

3) (-5xy) (4x2) (y4)

(-5) (4) (1) x3+1 y2+1

4) (-3j2k4) (2jk6)

(-3) (2) j2+1 k4+6

-6j3k10

Power to a Power

(x2)3 = x2 * x2 * x2 = x2+2+2 = x6

x x x x x x = x6

x2*3 = x6

RULE: Multiply exponents.

EXAMPLES:

1) (xy3)4 = xy3 * xy3 * xy3 * xy3 = x4 x12

= x1*4 y3*4 = x4 x12

4/11/12

Multiplying a Polynomial by a Monomial

RULE: Distribute!

EXAMPLES:

1) x(5x + x2)

x(5x) + x(x2)

5x2 + x3

2) -2xy(2xy + 4 x2)

(-2xy) (2y) + (-2xy) (4 x2)

-4x y2 + -8 x3y

3) 2x2y2(3xy + 2y + 5x)

6x3y3 + 4x2y3 + 10x3y2

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4/12/12

Binomials Times Binomials

There are 3 techniques you can use for multiplying polynomials:

1. Distributive Property

2. FOIL Method

3. The Box Method

EXAMPLE OF DISTRIBUTIVE PROPERTY OR FOIL METHOD:

(2x + 3) (5x + 8)

First: (2x)(5x) = 10x2

Outer: (2x)(8) = 16x

Inner: (3)(5x) = 15x

Last: (3)(8) = 24

Combine like terms: 10x2 + 31x + 24

EXAMPLES OF BOX METHOD:

(3x – 5) (5x + 2)

| |3x |-5 | |

|5x |15 x2 |-25x |15 x2 – 25x + 6x – 10 |

|2 |6x |-10 |15 x2 – 19x – 10 |

(2x – 5) (x2 – 5x + 4)

| |x2 |-5x |4 |2x3 - 10x2 - 5x2 + 8x + 25x - 20 |

|2x |2 x3 |-10 x2 |8x |2x3 - 15x2 + 33x - 20 |

|-5 |-5x2 |25x |-20 | |

4/13/12

Dividing Monomials

When dividing monomials, subtract the exponents

1. b5 = b * b * b * b * b = b5-2 = b3

b2 b * b

end

-----------------------

+ ? =

)

(

( find them by using the “Thumb Rule”

( This means x is 0, so this ordered pair will be (0, )

X

( This means y is 0, so this ordered pair will be ( , 0 )

X

( This means x is 0, so this ordered pair will be (0, )

X

( This means y is 0, so this ordered pair will be ( , 0 )

X

( This means x is 0, so this ordered pair will be (0, )

X

X

( This means y is 0, so this ordered pair will be ( , 0 )

Zero slope (m = 0)

negative

positive

Undefined (no slope)

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