Factoring polynomials completely
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The document provides instructions on how to completely factor polynomials. It gives examples of factoring expressions like 4n^2 - 16, 5x^2 - 45, and 3x^2 - 18x - 21. The steps explained are to find the greatest common factor, factor any binomials or trinomials, and simplify the expression as much as possible. Additional practice problems and their factored solutions are provided.
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Math083 day 1 chapter 6 2013 falljbianco9910
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This document provides information about a college algebra course taught by Mr. Bianco in Fall 2013. It covers topics like multiplying polynomials using FOIL, factoring polynomials by finding the greatest common factor and factoring trinomials. Examples are provided to demonstrate how to factor polynomials of different forms like ax^2 + bx + c. Students are reminded to check their factoring work by multiplying the factors.EPCA_MODULE-2.pptx
EPCA_MODULE-2.pptxBenCorejadoAgarcio
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To factor a polynomial using greatest common factors (GCF):
1. Find the GCF of the coefficients and of the variables.
2. The GCF is the factored form of the polynomial.
3. Checking the factored form using the distributive property verifies the correct factorization.Solving by factoring remediation notes
Solving by factoring remediation notesMichelle Barnhill
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Here are the key steps to factoring when a ≠1:
1. Check for GCF. If there is a GCF, factor it out.
2. If no GCF, find two numbers whose product is ac and sum is b. These will be the coefficients of the factors.
3. Split the middle term into two terms using the factors.
4. Factor using the box method, finding the GCF of each row and column.
Some examples:
2x2 + 11x + 5
ac = 10, factors are 2, 5
(2x + 1)(x + 5)
3x2 - 5x - 2
ac = -Factoring and Box Method
Factoring and Box Methodswartzje
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This document discusses various techniques for factoring polynomials, including:
1. Factoring using the greatest common factor (GCF).
2. Factoring polynomials with 4 or more terms by grouping.
3. Factoring trinomials using factors that add up to the coefficient of the middle term.
4. Using the "box method" to factor trinomials where the coefficient of the x^2 term is not 1.
5. Factoring the difference of two squares using the formula a^2 - b^2 = (a + b)(a - b).5.2 divede and conquer 03
5.2 divede and conquer 03Krish_ver2
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This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).5.2 divede and conquer 03
5.2 divede and conquer 03Krish_ver2
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This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).More from ListeningDaisy (11)
Geometry dots & lines
Geometry dots & linesListeningDaisy
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This document provides an overview of basic geometry concepts including:
- Points have no dimension and are named using capital letters
- A line has no end points, is straight, and goes off to infinity in two directions
- A line segment is part of a straight line with two distinct end points
- A ray has one endpoint and extends off to infinity in one direction
- Lines, line segments, and rays can intersect at a single point
- A plane is a flat, two-dimensional surface that can be named using points or a cursive letterStem & leaf plots + histograms
Stem & leaf plots + histogramsListeningDaisy
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This document provides instructions and examples for creating different types of data displays, including stem-and-leaf plots, histograms, back-to-back stem-and-leaf plots, and box-and-whisker plots. It includes examples using temperature data and adoption prices for dogs. Key terms and how to interpret the visuals are explained through mini lessons within the document.Measures of central tendency
Measures of central tendencyListeningDaisy
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This document discusses measures of central tendency and uses the scores of 28, 18, 24, 22, 18 to demonstrate how to calculate the median, mean, mode, and range. It defines median as the middle number when values are ordered from lowest to highest. Mean is calculated by adding all values and dividing by the total count. Mode is the number with the most occurrences. Range is found by subtracting the minimum from the maximum value.Factor by grouping
Factor by groupingListeningDaisy
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The document provides steps for factoring polynomial expressions:
1) Separate the expression into terms with common factors and group those terms;
2) Find the greatest common factor (GCF) of the grouped terms;
3) Enclose the GCF and the grouped terms in parentheses to arrive at the factored expression.
Worked examples demonstrate factoring various polynomials, including using negative signs properly.Factoring Special Products in Difference of Squares
Factoring Special Products in Difference of SquaresListeningDaisy
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The document provides examples of factoring quadratic expressions of the form a^2 +/- b^2 into the form (a +/- b)(a -/+ b). It demonstrates factoring several quadratic expressions using this method, including expressions with coefficients other than 1. It also introduces a formula for factoring expressions of the form a^2 + 2ab + b^2 and applies this formula to one example. Finally, it factors one expression where the middle term is negative by distributing the negative sign appropriately.Factoring Pt. 2/2 (x^2 + bx + c)
Factoring Pt. 2/2 (x^2 + bx + c)ListeningDaisy
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1. The document discusses factoring quadratic expressions using the greatest common factor (GCF) method.
2. It provides examples of factoring expressions such as 3x^2 + 8x + 4 by trying different pairs of binomials that multiply to the GCF and constant term.
3. The document also addresses factoring expressions with negative coefficients, explaining that you can factor the expression inside the parentheses and multiply the whole expression by -1.Factoring pt. 1/2
Factoring pt. 1/2ListeningDaisy
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1. The document provides step-by-step instructions for factoring quadratic expressions using reverse FOIL. It explains that the factors are determined by the constant term's factors and shows examples of factoring expressions with positive and negative coefficients.
2. Problems include factoring expressions into standard form, factoring expressions with variables, and solving a word problem involving the area of two shapes to determine the width of one. The document breaks problems down into clear steps and provides answers on subsequent slides.Solving polynomial equations in factored form
Solving polynomial equations in factored formListeningDaisy
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The document provides instructions for solving polynomial equations in factored form. It begins by explaining that to solve an equation like (x – 5)(x + 4) = 0, one should not use the FOIL method but rather split the equation into two separate problems that each equal zero: x – 5 = 0 and x + 4 = 0. It then works through several examples of solving factored polynomial equations by finding the values of x that make each factor equal to zero. The document also covers factoring out the greatest common factor from expressions.
Math vocab 1
Math vocab 1ListeningDaisy
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1. The document defines key math vocabulary terms including variable, constant, coefficient, leading coefficient, like terms, monomial, polynomial, binomial, and trinomial.
2. Examples are provided to demonstrate identifying variables, constants, coefficients, like terms, and determining if expressions are monomials, polynomials, binomials, or trinomials.
3. The degree of a polynomial is defined as the highest exponent present in the polynomial. Examples show determining the degree of various polynomials.Factoring polynomials completely
- 1. Factoring Polynomials Completely By L.D.
- 2. Problem 1 Instructions: Factor 4n2 – 16 completely.
- 3. Problem 1 Instructions: Factor 4n2 – 16 completely. The first thing we need to do here is to find the GCF and make the problem look like 4(n2 – 4)
- 4. Problem 1 Instructions: Factor 4n2 – 16 completely. The first thing we need to do here is to find the GCF and make the problem look like 4(n2 – 4) The next thing we need to do is to factor the binomial inside, n2 – 4 will become (n + 2)(n - 2).
- 5. Problem 1 Instructions: Factor 4n2 – 16 completely. The first thing we need to do here is to find the GCF and make the problem look like 4(n2 – 4) The next thing we need to do is to factor the binomial inside, n2 – 4 will become (n + 2)(n - 2). The last thing to be done is that the problem will end up 4(n + 2)(n - 2) which is factored as far as we can take it.
- 6. Problem 1 Extras Factor These Completely: - 5x2 - 45 - 4 – 30 - 8x2 + 64
- 7. Problem 1 Extras Factor These Completely: - 5x2 – 45 5(x + 3)(x – 3) - 8d3 + 24d 4d(2d2 + 6) - 8x2 + 64
- 8. Problem 2 Instructions: Factor 9y3 + 81y2 = 0 completely and solve
- 9. Problem 3 Instructions: Factor 3x2 – 18x – 21 completely.
- 10. Problem 3 Instructions: Factor 3x2 – 18x – 21 completely. First, find the GCF to get 3(x2 – 6x – 7)
- 11. Problem 3 Instructions: Factor 3x2 – 18x – 21 completely. First, find the GCF to get 3(x2 – 6x – 7) Now, you need to factor the interior to get (x + 7)(x – 1), making our answer 3(x + 7)(x – 1).