STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 2 nd Nine Weeks, 2016-2017 1
OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for students and parents. They have been revised this year as part of an internship process. Each nine weeks Standards of Learning (SOLs) have been identified and a detailed explanation of the specific SOL is provided. Specific notes have also been included in this document to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models for solving various types of problems. A section has also been developed to provide students with the opportunity to solve similar problems and check their answers. The answers to the document. problems are found at the end of the The document is a compilation of information found in the Virginia Department of Education (VDOE) Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE information, Prentice Hall textbook series and resources have been used. Finally, information from various websites is included. The websites are listed with the information as it appears in the document. Supplemental online information can be accessed by scanning QR codes throughout the document. These will take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the document to allow students to check their readiness for the nine-weeks test. The Algebra I Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of questions per reporting category, and the corresponding SOLs. Algebra I Blueprint Summary Table Reporting Categories No. of Items SOL Expressions & Operations 12 A.1 A.2a c A.3 Equations & Inequalities 18 A.4a f A.5a d A.6a b Functions & Statistics 20 A.7a f A.8 A.9 A.10 A.11 Total Number of Operational Items 50 Field-Test Items* 10 Total Number of Items 60 * These field-test items will not be used to compute the students scores on the test. It is the Mathematics Instructors desire that students and parents will use this document as a tool toward the students success on the end-of-year assessment. 2
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Laws of Exponents & Polynomial Operations A.2 The student will perform operations on polynomials, including a) applying the laws of exponents to perform operations on expressions; Monomial is a single term. It could refer to a number, a variable, or a product of a number and one or more variables. Some examples of monomials include: When you multiply monomials that have a common base, you add the exponents. Example 1: Multiply This works because when you raise a number or variable to a power, it is like multiplying it by itself that many times. When you then multiply this by another power of the same number or variable, you are just multiplying it by itself that many more times. Example 2:. Example 3: Simplify Scan this QR code to go to a video tutorial on multiplying monomials! When you raise a power to a power, you multiply the exponents. This means 3² times itself 4 times! Example 4: Simplify 4
Example 5: Simplify Often, you will be asked to multiply monomials and raise powers to a power. Make sure that you follow the ORDER OF OPERATIONS! Raise to powers first, then multiply. Example 6: Simplify Simplify each expression 1. 2. 3. 4. Laws of Exponents When you divide monomials with like bases, you will subtract the exponents. Anything raised to the zero power is equal to ONE! To find the power of a quotient, raise both the numerator and the denominator to the power. (Remember to follow the order of operations!) 5
Example 7: Scan this QR code to go to a video tutorial on dividing monomials! Example 8: You will also see negative exponents in monomials. When you have a negative exponent, you will reciprocate that variable (move it to the other side of the fraction bar) and the exponent will become positive. As an example: Remember that anything to the zero power equals 1! When simplifying monomials with negative exponents, you can start by flipping over all of the negative exponents to make them positive. Then, simplify. Example 9: Scan this QR code to go to a video tutorial on simplifying monomials with negative exponents! Example 10: 6
Exponents Laws of Exponents Simplify each expression 5. 6. 7. 8. 9. 10. Polynomials A.2 The student will perform operations on polynomials, including b) adding, subtracting, and multiplying polynomials. Adding and subtracting polynomials is the same as COMBINING LIKE TERMS. In order for two terms to be like terms, they must have the same variables and the same exponents. Like Terms NOT Like Terms Each of these terms contain an, therefore they are like terms. Although these terms have the same variables, corresponding variables do not have the same exponents. Therefore, these are NOT like terms. Example 1: Like terms are underlined here. Remember that each term takes the sign in front of it! 7
Remember that if you are subtracting a polynomial, you are subtracting all of the terms (Therefore, you must distribute the negative to each term first!) Example 2: Distribute the negative to everything in the second set of parentheses! Then, COMBINE LIKE TERMS! Polynomials Scan this QR code to go to a video tutorial on adding and subtracting polynomials. Simplify each expression 1. 2. 3. 4. 5. To multiply a polynomial by a monomial, simply distribute the monomial to each term in the polynomial. You will use the rules of exponents to simplify each term. Example 3: Distribute the to each term. Then, simplify each term Example 4: Don t forget to check for like terms! Scan this QR code to go to a video tutorial on multiplying monomials and polynomials. 8
To multiply two polynomials together, distribute each term in the first polynomial to each term in the second polynomial. When you are multiplying two binomials together this may be called FOIL. FOIL stands for: F First multiply the first term in each binomial together O Outer multiply the outermost term in each binomial together I Inner multiply the innermost term in each binomial together L Last multiply the last term in each binomial together (This is the exact same as distributing the first term, then distributing the second term) Don t forget to combine like terms when possible. Example 5: First Outer Inner Last Example 6: Example 7: Remember that to square something means to multiply it by itself! 6. 7. 8. 9. 10. Polynomials 9
Factoring A.2 The student will perform operations on polynomials, including c) factoring completely first- and second-degree binomials and trinomials in one or two variables. The prime factorization of a number or monomial is that number or monomial broken down into the product of its prime factors. Example 1: Write the prime factorization of 9 2 3 3 or To find the greatest common factor (GCF) of two or more monomials, break each down into its prime factorization. The GCF is the product of all of the shared factors. Example 2: What is the greatest common factor of Circle each factor that they ALL have in common! You can use the GCF to help you rewrite (factor) polynomials. If all of the terms in the polynomial have common factors you can pull these factors out from the terms to factor the polynomial. Example 3: Factor GCF = Pull the GCF out from each term and rewrite. Check your work by distributing. 10
Example 4: Factor GCF = Pull the GCF out from each term and rewrite. Check your work by distributing. Factoring Scan this QR code to go to a video tutorial on greatest common factors. 1. Write the prime factorization of 2. Find the greatest common factor of 3. Factor Simplifying Radicals A.3 The student will express the square roots and cube roots of whole numbers and monomial algebraic expressions in simplest radical form. To simplify a radical, you will pull out any perfect square factors (i.e. 4, 9, 16, 25, etc.) The square root of 9 is equal to 3, so you can pull the square root of 9 from underneath the radical sign to find the simplified answer, which means 3 times the square root of 2. You can check this simplification in your calculator by verifying that. Another way to simplify radicals, if you don t know the factors of a number, is to create a factor tree and break the number down to its prime factors. When you have broken the number down to all of its prime factors you can pull out pairs of factors for square roots, which will multiply together to make perfect squares. 11
Example 5: Simplify 2 64 8 8 4 2 2 4 Example 6: Simplify 2 2 2 2 16 2 x x x y 4 4 2 2 2 2 To simplify a root of a higher index, you pull out factors that occur the same number of times as the index of the radical. As an example, if you are simplifying, you would only pull out factors that occurred 5 times, since 5 is the index of the root. Example 7: Simplify Because this is a cube root, I pulled out things that occurred 3 times. Scan this QR code to go to a video tutorial on simplifying radicals. Factoring Simplify the following radicals. 4. 5. 6. 12
Factoring Special Cases A.2 The student will perform operations on polynomials, including c) factoring completely first- and second-degree binomials and trinomials in one or two variables. A.3 The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form. Factoring Trinomials To factor a trinomial of the form to, and whose product is equal to. You can start by listing all of the factors of to the coefficient of., first find two integers whose sum is equal, and then see which two factors add up Once you have determined which factors to use, you can put all of your terms in a box and factor the rows and columns. Example 1: Factor So, we are looking for factors of 8 that add up to 6! Factors of 8 Sum of factors 1, 8 9 2, 4 6 Find the greatest common factor in each row and each column. These will give you your two binomials! Put terms in a box First Term ( ) Other Factor One Factor Last Term (c) Check your answer by FOIL-ing! Sometimes you will not be able to find factors of happens, the trinomial is PRIME. that sum to b. When this 13
Example 2: Factor So, we are looking for factors of -4 that add up to 5! Factors of 8 Sum of factors 1, -4-3 -1, 4 3-2, 2 0 Nothing works, therefore this trinomial is PRIME When factoring, anytime the term is negative and the term is positive, your answer will have two minus signs! Example 3: Factor So, we are looking for factors of 80 that add up to! Factors of 80 Sum of factors -4, -20-24 -5, -16-21 Find the greatest common factor in each row and each column. These will give you your two binomials! Check your answer by FOIL-ing! Example 4: Factor Pull out a GCF first!! So, we are looking for factors of 15 that add up to! Find the greatest common factor in each row and each column. These will give you your two binomials! Check your answer by FOIL-ing! Don t forget your GCF in the front. 14
Factoring Special Cases Factor each of the trinomials below 1. 2. 3. 4. To solve a quadratic equation (i.e. find its solutions, roots, or zeros), set one side equal to zero (put the quadratic in standard form), then factor. Set each factor equal to zero to find the values for that are the solutions to the quadratic. Example 5: Find the zeros of Scan this QR code to go to a video tutorial on factoring trinomials. Start by getting one side equal to zero and write in standard form. Now factor the trinomial. We are looking for factors of that add up to. and work! Set both factors equal to zero! You can check your answer in your calculator by graphing the quadratic. The solutions are the x-intercepts, so this graph should cross the x-axis at -2 and 9. or 15-2 9
Factoring Special Cases Find the solution to each trinomial 5. 6. 7. Scan this QR code to go to a video tutorial on solving trinomials by factoring. Special Cases A perfect square trinomial can be factored to two binomials that are the same, so you can write it as the binomial squared. Example 6: Factor If your first and last terms are perfect squares you can check for a perfect square trinomial. Take the square root of the first and last number and see if the product of those is equal to ½ of the middle number. Now that we know this case works, you can write the binomial factor squared Remember to check your answer by FOIL-ing the binomials back out! Another special case is if the quadratic is represented as the difference of two perfect squares (i.e. ). If both the first and last term are perfect squares, and the two terms are being subtracted their factorization can be written as. As an example. Remember that you can check your work by FOIL-ing. Example 7: Factor completely To begin, you should factor out a GCF. In this case it would be 3. Now you are left with a difference of squares! 16
8. 9. 10. Factoring Special Cases Scan this QR code to go to a video tutorial on factoring special cases. Solving Quadratic Equations A.4 The student will solve multistep linear and quadratic equations in two variables c) solving quadratic equations algebraically and graphically; Graphing a quadratic equation Standard form for a quadratic function is: The graph of a quadratic equation will be a parabola. If, then the parabola opens upward. If, then the parabola opens downward. The axis of symmetry is the line. The x-coordinate of the vertex is plugging that x value into the equation and solving for The y-intercept is To graph a quadratic: 1. Identify a, b, and c. 2. Find the axis of symmetry ( ), and lightly sketch.. The y-coordinate of the vertex is found by 3. Find the vertex. The x-coordinate is. Use this to find the y-coordinate. 4. Plot the y-intercept (c), and its reflection across the axis of symmetry. 5. Draw a smooth curve through your points. The vertex of a parabola is its turning point, or the tip of the parabola. In this picture, the turning point is at (2, 0). Scan this QR code to go to a video tutorial on graphing and solving quadratic equations. 17
Example 1: Graph Step 1: Identify a, b, and c. Step 2: Find and sketch the axis of symmetry. Step 3: Find the vertex. The x-coordinate is 1. Plug this in to find y. The vertex is (1, 1). Step 4: Plot the y-intercept and its reflection. Because c = 3, the y-intercept is (0, 3). Reflecting this point across x = 1 gives the point (2, 3). Step 5: Draw a smooth curve. Remember to check your graphs in your calculator! You might be asked to find the solutions of a quadratic equation by graphing it. The solutions to a quadratic equation are the points where it crosses the x-axis. A quadratic can have two solutions, only one solution, or no solutions at all. Two Solutions (the parabola crosses the x-axis twice) One Solution (the parabola crosses the x-axis one time) 18 No Solutions (the parabola does not cross the x- axis)
Sometimes you will need to find the solution to a quadratic that cannot be factored. In that case, you can use the quadratic formula: You just substitute the values for a, b, and c into the quadratic formula and simplify. Example 2: Solve Plug these values into the quadratic formula Your two solutions are and Scan this QR code to go to a video tutorial on using the quadratic formula. Solving Quadratic Equations 1. Sketch the graph of 2. Sketch the graph of 3. Find the solution(s) by graphing 4. Find the solution(s) by graphing 5. Find the solution(s), use the quadratic formula = 0 6. Find the zero(s) of the quadratic, use any method you like. 19
Answers to the Laws of Exponents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Polynomials 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. problems: Factoring Special Cases 1. 2. 3. 4. Prime 5. or 6. or 7. or 8. 9. or 10. Solving Quadratic Equations 1. 2. Factoring & Simplifying Radicals 1. 2. 3. 4. 5. 6. 3. or 4. or 5. 6. No Solution 20