Section 13.3 factoing trinomials by grouping
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This document discusses factoring trinomials by grouping. It provides examples of factoring trinomials where the coefficient of the second term is not 1, such as 3x^2 + 14x + 8 = (3x + 2)(x + 4). The document shows how to determine the sum and product of the terms, then find two numbers with that sum and product to group the terms. It provides step-by-step workings for factoring trinomials like 10n^2 + 3n - 1 and 20g^2 - g - 1 using this method. The last example factors the trinomial 12x^5 + 10x^4 - 12x^3 in two steps.
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The document discusses direct and inverse variation problems. It provides examples of how to solve each type of problem. For a direct variation problem, a constant k is determined using given values for variables x and y. Then k is used to find the value of y when a new value of x is given. For an inverse variation problem, a constant k is similarly determined using given values for variables that have an inverse relationship, like supply (S) and demand (D). Then k is used to find the value of the variable when a new value of the related variable is given.
Section 14.1 The fundamental property of rational expressions
This document discusses rational expressions and their applications. It covers objectives like finding the numerical value of a rational expression, determining values that make a rational expression undefined, writing rational expressions in lowest terms, and recognizing equivalent forms. Some key points include the definition of a rational expression as P/Q where P and Q are polynomials and Q ≠ 0, setting the denominator equal to 0 to find undefined values, and using the fundamental property of rational expressions to write expressions in lowest terms by dividing out common factors. Examples are provided for each concept.
Mat 092 section 12.3 scientific notation
This document discusses scientific notation. It begins by defining scientific notation as a way for scientists to express extremely large or small numbers using exponents. It then provides examples of writing numbers in scientific notation by positioning the decimal point and determining the appropriate exponent. The document also demonstrates how to convert numbers written in scientific notation to a standard numerical form without exponents. Finally, it shows how to perform calculations using numbers expressed in scientific notation.
Mat 092 section 12.2 integer exponents
This document discusses rules and concepts related to exponents and polynomials, including:
1) Using 0 as an exponent, where a0 = 1 for any nonzero real number a.
2) Using negative exponents, where an = 1/an.
3) The quotient rule for exponents, where (am/an) = am-n.
4) Combining these rules to simplify expressions involving exponents. Examples are provided to illustrate each concept and rule.
15.3 solving systems of equations by elimination
The document discusses solving systems of linear equations by elimination. It covers:
1) Using the elimination method to solve systems by eliminating one variable and finding the value of the other variable.
2) Multiplying equations when necessary to make coefficients of one variable opposite to eliminate that variable.
3) Using an alternative method like eliminating the other variable to find the second solution value if the first method is messy.
15.2 solving systems of equations by substitution
This document provides examples for solving systems of linear equations by substitution. It begins with an example showing the step-by-step process for solving a system using substitution. Subsequent examples demonstrate how to solve special systems that have no solution, infinitely many solutions, and systems with fractions or decimals. The final example walks through solving a system with fractions using clear fractions before substitution.
15.1 solving systems of equations by graphing
The document discusses solving systems of linear equations and inequalities through graphing. It presents examples of determining if an ordered pair is a solution to a system, graphing systems on coordinate planes to find their point of intersection as the solution, and identifying special cases where systems have no solution or an infinite number of solutions. Three key cases are described: 1) a single solution where graphs intersect at one point, 2) no solution where graphs are parallel lines, and 3) infinite solutions where graphs are the same line.
11.6 graphing linear inequalities in two variables
This document discusses how to graph linear inequalities in two variables. It provides examples of graphing various inequalities, including 2x + 3y ≤ 6, 2x + y > 3, and y < 3x. The process involves graphing the boundary line defined by the equal case of the inequality and then shading the appropriate region based on checking if a test point satisfies the inequality. Key steps are to draw the boundary line as solid or dashed based on the symbols (< or > versus ≤ or ≥) and then shade the region containing points that satisfy the inequality.
11.5 point slope form of a linear equation
The document discusses writing equations of lines using different forms. It explains how to use point-slope form to write an equation of a line given a point and slope. It also shows how to write an equation of a line using two points on the line. Finally, it demonstrates how to write an equation that models a linear data set by plotting the data points and using two points and the point-slope form.
11.4 slope intercept form of a linear equation
This document discusses graphing and writing equations of lines in slope-intercept form. It covers identifying the slope and y-intercept from an equation in slope-intercept form, graphing a line by using its slope and a given point, writing an equation of a line given its slope and a point, and graphing horizontal and vertical lines. Examples are provided to illustrate finding slope and y-intercept, graphing lines by plotting points, writing equations, and graphing horizontal and vertical lines using a given point.
11.3 slope of a line
This document discusses finding the slope of a line from two points or an equation. It provides the slope formula and explains how to calculate slope given two points on a line. It also discusses horizontal and vertical lines, which have slopes of 0 and undefined, respectively. The document shows how to find the slope of a line from its equation by solving for y and taking the coefficient of x. It concludes by explaining how to determine if two lines are parallel, perpendicular, or neither based on the equality or product of their slopes. Examples are provided to demonstrate these concepts.
11.2 graphing linear equations in two variables
The document discusses how to graph linear equations and inequalities in two variables. It provides examples of graphing linear equations by plotting ordered pairs, finding intercepts, and using linear equations to model data. Specifically, it shows how to graph equations of the form y=mx+b, Ax+By=0, y=b, and x=a. It demonstrates finding intercepts and using them to graph equations. Finally, it gives an example of using a linear equation to model the monthly costs of a small business based on the number of products sold.
11.1 linear equations in two variables
This document discusses graphs of linear equations and inequalities in two variables. It covers interpreting graphs, writing solutions as ordered pairs, deciding if an ordered pair is a solution to an equation, completing ordered pairs, completing tables of values, and plotting ordered pairs on a coordinate plane. The objectives are to be able to perform each of these tasks related to linear equations in two variables represented in rectangular coordinate systems.
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Section 13.3 factoing trinomials by grouping
- 1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G 2 Factoring and Applications 13
- 2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G 1. Factor trinomials by grouping when the coefficient of the second-degree term is not 1. Objectives 13.3 Factoring Trinomials by Grouping
- 3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Factor Trinomials by Grouping Factor the trinomial 3x2 + 14x + 8. Sum is 14. Product is 24. 1 24 2 12 3 8 4 6 Find a product of 24 and a sum of 14. 3x + x + x + 82 2 12 x (3x + 2) + 4 (3x + 2) (3x + 2) (x + 4) (3x + 2) (x + 4) =Check: 3x2 + 14x + 8 Example 2 3 14 8x x
- 4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Factor Trinomials by Grouping Factor the trinomial 10n2 + 3n – 1. 10n2 + 3n – 1 Sum is 3. Product is – 10. 1 –10 2 –5 Find a product of –10 and a sum of 3. 10n2 + n – n – 15 2 5n (2n + 1) – 1 (2n + 1) (2n + 1) (5n – 1) (2n + 1) (5n – 1) =Check: 10n2 + 3n – 1 –1 10 –2 5 Example
- 5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Factor Trinomials by Grouping Factor the trinomial 20g2 – g – 1. Sum is –1. Product is – 20. 1 –20 2 –10 4 –5 Find a product of –20 and a sum of –1. 20g2 + g – g – 14 5 4g (5g + 1) – 1 (5g + 1) (5g + 1) ( 4g – 1) 20g2 – g – 1 Check: (5g + 1) (4g – 1) = 20g2 – g – 1 Example
- 6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Factor Trinomials by Grouping –1 36 –2 18 –3 12 Find a product of – 36 and a sum of 5. Factor 12x5 + 10x4 – 12x3 . 12x5 + 10x4 – 12x3 2x3 (6x2 + 5x – 6)= 2x3 = 2(6x + x – x – 6) –4 9 –6 6 9 4 2x3 = (3x ( 2x + 3) – 2 (2x + 3)) 2x3= ((2x + 3) ( 3x – 2)) 2x3= (2x + 3) (3x – 2) Example
- 7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Factor Trinomials by Grouping Factor 12x + 10x – 12x .5 4 3 Check: 32x ( 2x + 3 ) ( 3x – 2 ) 4( 4x ( 3x – 2 )+ 6x )3 5 12x – 8x4 = = + 18x4 – 12x3 = 12x + 10x – 12x5 4 3 Example (cont)