Slide 1 / 276 Slide 2 / 276 Algebra II Polynomials: Operations and Functions 2014-10-22 www.njctl.org Slide 3 / 276 Slide 4 / 276 click on the topic to go Table of Contents to that section Properties of Exponents Review Operations with Polynomials Review Special Binomial Products Properties of Binomial Theorem Exponents Review Factoring Polynomials Review Dividing Polynomials Return to Table of Polynomial Functions Contents Analyzing Graphs and Tables of Polynomial Functions Zeros and Roots of a Polynomial Function This section is intended to be a brief review of this topic. Writing Polynomials from its Given Zeros For more detailed lessons and practice see Algebra 1. Slide 5 / 276 Slide 6 / 276 Goals and Objectives Why do we need this? · Students will be able to simplify complex Exponents allow us to condense bigger expressions containing exponents. expressions into smaller ones. Combining all properties of powers together, we can easily take a complicated expression and make it simpler.
Slide 7 / 276 Slide 8 / 276 Properties of Exponents Product of Powers Power of Powers Power of a product Negative exponent Power of 0 Quotient of Powers Slide 9 / 276 Slide 10 / 276 . 1 Simplify: 5m 2 q 3 10m 4 q 5 A 50m 6 q 8 B 15m 6 q 8 C 50m 8 q 15 D Solution not shown Slide 11 / 276 Slide 12 / 276 3 Divide: 4 Simplify: A C C A B D Solution not B D Solution not shown shown
Slide 13 / 276 Slide 14 / 276 Sometimes it is more appropriate to leave answers with positive 5 Simplify. The answer may be in either exponents, and other times, it is better to leave answers without form. fractions. You need to be able to translate expressions into either form. Write with positive exponents: Write without a fraction: A C B D Solution not shown Slide 15 / 276 Slide 16 / 276 When fractions are to a negative power, a short-cut is to invert the fraction and make the exponent positive. 6 Simplify and write with positive exponents: Try... A C B D Solution not shown Slide 17 / 276 Slide 18 / 276 7 Simplify and write with positive exponents: Two more examples. Leave your answers with positive exponents. A C B D Solution not shown
Slide 19 / 276 Slide 20 / 276 Operations with Polynomials Review Return to Table of Contents This section is intended to be a brief review of this topic. For more detailed lessons and practice see Algebra 1. Slide 21 / 276 Slide 22 / 276 Vocabulary Review A monomial is an expression that is a number, a variable, or the Goals and Objectives product of a number and one or more variables with whole number exponents. · Students will be able to combine polynomial A polynomial is the sum of one or more monomials, each of functions using operations of addition, which is a term of the polynomial. subtraction, multiplication, and division. Put a circle around each term: Slide 23 / 276 Slide 24 / 276 Polynomials can be classified by the number of terms . The table below summarizes these classifications.
Slide 25 / 276 Slide 26 / 276 Identify the degree of each polynomial: Polynomials can also be classified by degree . The table below summarizes these classifications. Slide 27 / 276 Slide 28 / 276 Drag each relation to the correct box: Polynomial Function Polynomial Functions Not Polynomial Functions A polynomial function is a function in the form where n is a nonnegative integer and the coefficients are real numbers. The coefficient of the first term, a n , is the leading coefficient . A polynomial function is in standard form when the terms are in order of degree from highest to lowest. f(x) = For extra practice, make up a few of your own! Slide 29 / 276 Slide 30 / 276 To add or subtract polynomials, simply distribute the + or - sign to Closure : A set is closed under an operation if when any two each term in parentheses, and then combine the like terms from elements are combined with that operation, the result is also each polynomial. an element of the set. Examples: (2a 2 +3a - 9) + (a 2 - 6a +3) Is the set of all polynomials closed under - addition? (2a 2 +3a - 9) - (a 2 - 6a +3) - subtraction? Watch your signs...forgetting to distribute the minus sign is one Explain or justify your answer. of the most common mistakes students make !!
Slide 31 / 276 Slide 32 / 276 9 Simplify A B C D Slide 33 / 276 Slide 34 / 276 12 What is the perimeter of the following figure? (answers are in units, assume all angles are right) 2x - 3 A x 2 +5x - 2 B - 10x + 1 C 8x 2 - 3x + 4 D Slide 35 / 276 Slide 36 / 276 Multiplying Polynomials 13 What is the area of the rectangle shown? To multiply a polynomial by a monomial, you use the distributive property of multiplication over addition together with the A laws of exponents. B C Example: Simplify. D -2x(5x 2 - 6x + 8) (-2x)(5x 2 ) + (-2x)(-6x) + (-2x)(8) -10x 3 + 12x 2 + -16x -10x 3 + 12x 2 - 16x
Slide 37 / 276 Slide 38 / 276 14 15 Find the area of a triangle (A= 1 / 2 bh) with a base of 5 y and a height of 2 y + 2 . All answers are in square units. A A B B C C D D Slide 39 / 276 Slide 40 / 276 Compare multiplication of polynomials with multiplication of integers. How are they alike and how are they different? Discuss how we could check this result. = Is the set of polynomials closed under multiplication? Slide 41 / 276 Slide 42 / 276 To multiply a polynomial by a polynomial, distribute each term of 16 What is the total area of the rectangles shown? the first polynomial to each term of the second. Then, add like terms. A Before combining like terms, how many terms will there be in each product below? B C 3 terms x 5 terms D 5 terms x 8 terms 100 terms x 99 terms
Slide 43 / 276 Slide 44 / 276 17 18 A A B B C C D D Slide 45 / 276 Slide 46 / 276 Example Part A: A town council plans to build a public parking lot. The outline below represents the proposed shape of the parking lot. Write an expression for the area, in square yards, of this proposed parking lot. Explain the reasoning you used to find the expression. From High School CCSS Flip Book Slide 47 / 276 Slide 48 / 276 Example Part B: Example Part C: The town council has plans to double the area of the parking lot in a The town council’s second plan to double the area changes the few years. They create two plans to do this. The first plan increases shape of the parking lot to a rectangle, as shown in the diagram the length of the base of the parking lot by p yards, as shown in the below. diagram below. Write an expression in terms of x to represent the value of p , in feet. Explain the reasoning you used to find the value Can the value of z be represented as a polynomial with integer of p . coefficients? Justify your reasoning.
Slide 49 / 276 Slide 50 / 276 20 Find the value of the constant a such that A 2 B 4 C 6 Special Binomial Products D -6 Return to Table of Contents Slide 51 / 276 Slide 52 / 276 Square of a Sum Square of a Difference (a + b) 2 = (a - b) 2 = (a + b)(a + b) = (a - b)(a - b) = a 2 + 2ab + b 2 a 2 - 2ab + b 2 The square of a + b is the square of a plus twice the product of a and b The square of a - b is the square of a minus twice the product of plus the square of b. a and b plus the square of b. Example: Example: Slide 53 / 276 Slide 54 / 276 2 Product of a Sum and a Difference + = (a + b)(a - b) = a 2 + -ab + ab + -b 2 = Notice the sum of -ab and ab a 2 - b 2 equals 0. 2 2 The product of a + b and a - b is the square of a minus the +2 + square of b. Example: Practice the square of a sum by putting any monomials in for and .
Slide 55 / 276 Slide 56 / 276 2 - = + - = 2 2 2 2 - - 2 + This very important product is called the difference of squares. Practice the square of a difference by putting any monomials in for and . How does this problem differ from the last? Practice the product of a sum and a difference by putting any Study and memorize the patterns!! You will see them over and monomials in for and . How does this problem differ over again in many different ways. from the last two? Slide 57 / 276 Slide 58 / 276 21 22 Simplify: A A B B C D C D Slide 59 / 276 Slide 60 / 276 23 Simplify: 24 Multiply: A A B B C C D D
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