27 Wyner Math 2 Spring 2019

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1 27 Wyner Math 2 Spring 2019 CHAPTER SIX: POLYNOMIALS Review January 25 Test February 8 Thorough understanding and fluency of the concepts and methods in this chapter is a cornerstone to success in the rest of this course and most future math courses as well. 6-A Properties of Exponents Monday 1/7 scientific notation ➊ Simplify an expression using properties of exponents. ➋ Convert calculator notation to scientific notation and to standard notation. ➌ Convert standard notation to scientific notation. 6-B Algebraic Expressions Monday 1/14 term expression argument polynomial monomial binomial trinomial degree constant linear quadratic cubic standard form coefficient leading coefficient ➊ Identify terms of an expression. ➋ Identify arguments of functions. ➌ Simplify a fraction with multiple terms in the numerator or denominator. ➍ Multiply a rational expression by an integer. ➎ Classify a polynomial in one variable. 6-C Addition, Subtraction, and Multiplication Wednesday 1/16 conjugate ➊ Add or subtract polynomials. ➋ Multiply two polynomials. ➌ Multiply more than two polynomials. ➍ Multiply a binomial by its conjugate. ➎ Square a binomial. 6-D Factoring Friday 1/18 factor common monomial ➊ Factor a trinomial by guessing and checking. ➋ Factor a common monomial out of each term of a polynomial. ➌ Factor a polynomial by grouping. ➍ Factor ax 2 + bx + c by grouping. ➎ Factor a perfect square trinomial, a difference of two squares, a difference of two cubes, or a sum of two cubes. ➏ Factor any polynomial. 6-E Solving by Factoring Friday 1/25 ➊ Solve an equation in factored form. ➋ Solve an equation by factoring.

2 28 Wyner Math 2 Spring A Properties of Exponents The rules below are valid in almost all contexts, including any time a and b are positive or x and y are integers. Property Rule Example Power of a Product (ab) x = a x b x (2x) 3 = 2 3 x 3 = 8x 3 Power of a Quotient ( a b) x = ax b x (x 2) 3 = 2 x3 8 Power of a Power (b x ) y = b xy (x 5 ) 3 = x 15 Product of Powers b x b y = b x+y x 5 x 3 = x 8 b Quotient of Powers b = x 5 y bx-y x = 3 x2 Zero Exponent b 0 = = 1 Negative Exponent b -x = b 1 x 2-3 = 2 1 = ➊ Simplify an expression using properties of exponents. 1. Use the properties above as needed. 2. Reduce fractions if needed. ➊ Simplify. a) ab 3 (a 2 b) 5 b) ( 5a 2 ) 3 c) ( 5a 2 ) -3 d) 10( 5a 2 ) -3 e) 8a-3 6a 4 f) ab 3 (a 10 b 5 125a ) 8 ( 5a) ( 5a) 2 3 3a a 11 b (8) 125a 3 125a a a 3 ➊ Simplify (2a4 b) 3 b 6, and write it without a fraction. State each property of exponents used. a 12 bc 2 (2a 4 b) 3 = 2 3 (a 4 ) 3 b 3 Power of a Product (a 4 ) 3 = a 12 Power of a Power b 3 b 6 = b 9 a 12 b 9 a 12 b = a 0 b 8 a 0 = 1 Product of Powers Quotient of Powers Zero Exponent 8(a 4 ) 3 b 3 b 6 a 12 bc 2 8a 12 b 3 b 6 a 12 bc 2 8a 12 b 9 a 12 bc 2 8a 0 b 8 c 2 8(1)b 8 c 2 1 c 2 = c-2 Negative Exponent 8b 8 c -2

3 29 Wyner Math 2 Spring 2019 SCIENTIFIC NOTATION is a 10 b, where 1 a < 10 and b is an integer. Many calculators use the notation aeb instead of a 10 b to display scientific notation. Do not write numbers in this notation. ➋ Convert calculator notation to scientific notation and to standard notation. 1. To convert aeb to scientific notation, change E into 10, and make b an exponent. 2. To convert a 10 b to standard notation, move the decimal point right b spaces (which will be left if b is negative). Fill in 0 s as needed. ➋ a) 2.57E3 b) 2.57E ➌ Convert standard notation to scientific notation. 1. Move the decimal place left so that it follows the first nonzero digit, and count how many spaces b it was moved left. 2. Drop any zeros on the end, unless you know they were actually measured, not rounded. If the number is a decimal, no zeros were rounded. 3. Multiply this number by 10 b. ➌ a) 2700 b) 2700 exactly c) (b = 3) (b = 3) (b = -3)

4 30 Wyner Math 2 Spring B Algebraic Expressions A TERM is the product of any number of constants, variables, and function values. For example, 5, 5x 3, and 5x 3 yz 2 sin x are all terms. An Algebraic EXPRESSION is one or more terms added together. For example, 5, 5 + 5x 3, and 5 + 5x 3 5x 3 yz 2 sin x are all expressions. ➊ Identify terms of an expression. 1. Each individual term includes no addition or subtraction, but when the terms of an expression are added the result is the whole expression. ➊ Identify the terms of the expression 5x 2 8x x cos 3x. 5x 2, -8x 7, 9, and 10 x cos 3x are the terms of the expression, because when added the result is the whole expression 5x 2 8x x cos 3x. Note that 10, x, 3x, and -cos 3x are not terms of the whole expression.

5 31 Wyner Math 2 Spring 2019 An ARGUMENT of a Function is an expression input into the function. ➋ Identify arguments of functions. 1. An argument is a value input into a function. ➋ Identify all arguments in the expression 5x 2 8x x cos 3x. 3x is the argument of the cosine function. x cos 3x is the argument of the square root function. A function and its argument is a single term. An operation on such a term affects the term as a whole, not the argument separately. ➌ Simplify a fraction with multiple terms in the numerator or denominator. 1. Identify the terms of the numerator and the denominator. 2. Identify a factor that divides evenly into all of the terms. 3. Rewrite the fraction with the factor divided out of each term. If it is variable, specify that it cannot be zero. ➌ 6x2 9x 30x 6x 2 9x 1. The terms are 6x 2 and -9x 30x in the numerator, and 6x 2 and -9x in the denominator. 3x is an argument of the square root function, not a term of the numerator. 2. 6x 2 3x = 2x -9x 30x 3x = -3 30x (not -3 10) 6x 2 3x = 2x -9x 3x = x 3 30x, x 0 2x 3 Note that 6x 2 in the numerator and 6x 2 in the denominator do not cancel each other out because there are other terms. ➍ Multiply a rational expression by an integer. 1. Multiply each term in the numerator by the integer, making sure to avoid the following: multiplying a term more than once multiplying an argument multiplying a term in the denominator 11(4x) 10x ➍ Identify the error, if any, in each of the following attemps to multiply 5 by 2. 22(8x) 2 10x a) 5 The term 11(3x) should be multiplied by 2 one time, but it was multiplied by 2 once on the 11 and again on the 3x. 22(4x) 20x b) 5 10x is an argument and should not be multiplied. 22(4x) 2 10x c) 10 The denominator should not be multiplied. 22(4x) 2 10x d) 5 This is correct, because each term in the numerator was multiplied by 2 one time.

6 32 Wyner Math 2 Spring 2019 A MONOMIAL is a nonzero term with a whole number exponent. The exponent is the DEGREE. Degree Name Example 0 CONSTANT 5 (that is, 5x 0 ) 1 LINEAR 5x 2 QUADRATIC 5x 2 3 CUBIC 5x 3 n n th degree 5x 12 A POLYNOMIAL is an expression of the sum of one or more monomials, after like terms are combined. The degree of a polynomial in one variable is the highest degree of its terms. A BINOMIAL is a polynomial with two terms. A TRINOMIAL is a polynomial with three terms. A polynomial in one variable with its terms written in order from highest degree to lowest is in STANDARD Form. A COEFFICIENT is a constant value that is multiplied by a variable value. If a variable value is in a numerator, the coefficient is the rest of the fraction. The LEADING Coefficient of a Polynomial is the coefficient of the term with the highest degree. ➎ Classify a polynomial in one variable. 1. A single term is a monomial, two terms is a binomial, and three terms is a trinomial. 2. Consider the term with the highest exponent: If there is no variable (that is, an unwritten variable with an exponent of 0), the polynomial is a constant. If there is a variable with no exponent (that is, an exponent of 1), the polynomial is linear. If the variable is to the second power, the polynomial is quadratic. If the variable is to the third power, the polynomial is cubic. If the variable is to the fourth power or higher, look up what it is called, or simply refer to the polynomial as n th degree, where n is the exponent. ➎ Write the following polynomials in standard form, classify them, and identify the leading coefficient. a) x + 4x 3 b) -15x c) 8x 2 2x d) 2 7x x 2 + x standard form: 4x 3 + x -15x -2x 9 + 8x x x 2 + x + 2 classification: cubic binomial linear monomial 9 th degree trinomial cubic polynomial leading coefficient:

7 33 Wyner Math 2 Spring C Addition, Subtraction, and Multiplication ➊ Add or subtract polynomials. 1. Distribute any coefficient, including negatives. 2. Combine like terms. ➊ 5(4x 2 + 9x 3) (11x 4) 1. (20x x 15) + (-11x + 4) 2. 20x x 11 ➋ Multiply two polynomials. 1. Multiply each term in the first polynomial by each term in the second polynomial. 2. Simplify. 3. Combine like terms. ➋ (4x 2 3x) (x + 5) 1. (4x 2 x) + (4x 2 5) + (-3x x) + (-3x 5) 2. 4x x 2 3x 2 15x 3. 4x x 2 15x ➌ Multiply more than two polynomials. 1. Multiply the first two polynomials together (see ➋). 2. Multiply the result by the next polynomial. 3. Combine like terms. ➌ (x + 2) (x + 5) (x 10) 1. (x 2 + 7x + 10) (x 10) 2. (x 3 + 7x x) + (-10x 2 70x 100) 3. x 3 3x 2 60x 100

8 34 Wyner Math 2 Spring 2019 The CONJUGATE of a Binomial is the original binomial except with the sign of one of the terms switched: The conjugate of a + b is a b. There is a special pattern for multiplying a binomial by itself and for multiplying by its conjugate. Binomial times itself: (a + b) 2 = a 2 + 2ab + b 2 Binomial times its conjugate: (a + b)(a b) = a 2 b 2 ➍ Multiply a binomial by its conjugate. 1. Square each term of the binomial. 2. Subtract the second square from the first. ➍ (3x 10)(3x + 10) 1. a 2 = (3x) 2 = 9x 2 b 2 = 10 2 = (3x 10)(3x + 10) = 9x ➎ Square a binomial. 1. Square each term of the binomial. 2. Multiply the two terms together, and double this product. 3. Add the the three results of the steps above. ➎ (3x 10) 2 1. a 2 = (3x) 2 = 9x 2 b 2 = (-10) 2 = ab = 2(3x)(-10) = -60x 3. (3x 10) 2 = 9x 2 60x + 100

9 35 Wyner Math 2 Spring D Factoring Just as a factor of a whole number is a whole number that divides evenly into it, a FACTOR of a Polynomial is a polynomial that divides evenly into it. For example, 2 and 7 are factors of 14, and (x + 5) and (x 5) are factors of x Factoring a polynomial is writing it as a product of other polynomials. ➊ Factor a trinomial by guessing and checking. 1. Choose two terms that multiply together to equal the first term in the trinomial. 2. Choose two terms that multiply together to equal the last term in the trinomial. 3. Create two binomials from your four terms. 4. Multiply the binomials together to see if you get the desired polynomial. If not, start over with new terms. ➊ 12x 2 4x 5 retry: 1. 6x and 2x 6x and 2x 2. 5 and -1-5 and 1 3. (6x + 5) and (2x 1) (6x 5) and (2x + 1) 4. (6x + 5) (2x 1) = 12x 2 + 4x + 5 X (6x 5) (2x + 1) = 12x 2 4x 5

10 36 Wyner Math 2 Spring 2019 A COMMON Monomial is a term that divides evenly into every term in a polynomial. ➋ Factor a common monomial out of each term of a polynomial. 1. Find a monomial that divides evenly into every term in the polynomial. 2. Divide every term by the common monomial. 3. Write the common monomial, followed by the new polynomial in parentheses. 4. Repeat if possible. ➋ 40x 5 8x x x 2 divides into all three terms. 2. (40x 5 8x x 2 ) 4x 2 = 10x 3 2x x 2 (10x 3 2x + 5) Sometimes one monomial is common to some of the terms in a polynomial, and another monomial is common to the other terms. If factoring these separate monomials out of their respective terms leaves the same quotient each time, this grouping can help in factoring the polynomial. ➌ Factor a polynomial by grouping. 1. Sort the terms of the polynomial into groups such that each group has its own common monomial. 2. Factor a common monomial out of each group. 3. If the quotient of each group is the same, the factors of the polynomial are this quotient and the sum of the common monomials. If not, start over with different groupings or different monomials. ➌ 2x 3 8x 2 + 5x (2x 3 8x 2 ) + (5x 20) 2. 2x 2 (x 4) + 5(x 4) 3. (2x 2 + 5)(x 4) A factorable trinomial ax 2 + bx + c can be factored by grouping after splitting up bx into b 1 x + b 2 x, where b 1 b 2 = ac. ➍ Factor ax 2 + bx + c by grouping. 1. Calculate ac. 2. Identify a pair of factors of ac whose sum is b. 3. Using this pair of factors, rewrite bx as the sum of two separate terms. 4. Factor by grouping (see ➌). ➍ 12x 2 4x 5 1. ac = 12(-5) = (6) = -60, and = x 2 10x + 6x x(6x 5) + 1(6x 5) (2x + 1)(6x 5)

11 37 Wyner Math 2 Spring 2019 The following special cases are easy to factor when recognized. Type of Polynomial Polynomial Factors Perfect Square Trinomial a 2 + 2ab + b 2 (a + b) 2 Difference of Two Squares a 2 b 2 (a + b) (a b) Difference of Two Cubes a 3 b 3 (a b) (a 2 + ab + b 2 ) Sum of Two Cubes a 3 + b 3 (a + b) (a 2 ab + b 2 ) ➎ Factor a perfect square trinomial, a difference of two squares, a difference of two cubes, or a sum of two cubes. 1. Disregarding negatives, let a be the square root or cube root of the first term and let b be the square root or cube root of the last term. 2. See which of the four polynomials in the second column above matches the polynomial you are factoring. 3. Fill in a and b in the factorization shown in the third column above. 4. For a possible perfect square, give b the same sign as the middle term of the polynomial, and multiply the factors together to see if they do in fact result in the desired polynomial. ➎ Use special cases to factor the following polynomials if possible. Polynomial a b Type of Polynomial Factors a) x 2 100y 2 x 10y difference of two squares (x + 10y) (x 10y) b) x y 4 x 5 10y 2 difference of two squares (x y 2 ) (x 5 10y 2 ) c) x y 3 x 10y difference of two cubes (x 10y) (x 2 10xy + 100y 2 ) d) 27m 3 + 8p 3 3m 2p sum of two cubes (3m + 2p) (9m 2 6mp + 4p 2 ) e) 4x 2 20x x -5 perfect square (2x 5) 2 f) 4x 2 10x x -5 not a perfect square, because (2x 5) 2 = 4x 2 20x + 25, not 4x 2 10x + 25 ➏ Factor any polynomial. 1. If possible, factor out a common monomial from every term (see ➋). 2. If possible, factor a perfect square, a difference of squares or of cubes, or a sum of cubes (see ➍). 3. If the polynomial is not yet completely factored, try factoring by grouping (see ➌) or by guessing and checking (see ➊). ➏ 80x 7 180x x 5 (4x 2 9) 2. 20x 5 (2x + 3) (2x 3)

12 38 Wyner Math 2 Spring E Solving by Factoring If ab = 0, then a or b must be zero. ➊ Solve an equation in factored form. 1. Isolate zero on one side of the equation. If this cannot be done, the equation cannot be solved by this method. 2. Set each factor equal to zero. 3. Solve each equation in step 2. ➊ 3x 2 (x 5)(x + 6)(2x 9)(x 2 16) = x 2 = 0 x 5 = 0 x + 6 = 0 2x 9 = 0 x 2 16 = 0 2. x = 0 x = 5 x = -6 x = 9 2 x = ±4 ➋ Solve an equation by factoring. 1. Isolate zero on one side of the equation. If this cannot be done, the equation cannot be solved by this method. 2. Factor the expression. 3. Set each factor equal to zero. 4. Solve each equation in step 3. ➋ 18x 3 = 50x 1. 18x 3 50x = x (9x 2 25)= 0 2x (3x + 5) (3x 5) = x = 0 3x + 5 = 0 3x 5 = 0 4. x = 0 x = x = 5 3

Chapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring

Chapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Chapter Six Polynomials Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Properties of Exponents The properties below form the basis