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Algebra 1 Key Vocabulary Words Chapter 9 Section 9.1: monomial degree of monomial polynomial degree of polynomial leading coefficient binomial trinomial Section 9.2: algebra unit-tile (not a Key Vocabulary) algebra x-tile (not a Key Vocabulary) algebra x2-tile (not a Key Vocabulary) distributive property of multiplication (see Ch 2.5) FOIL pattern area of rectangle (not a Key Vocabulary) square of a binomial difference of two squares (sum and difference pattern) Section 9.3: perfect square trinomial difference of two squares Section 9.4: roots solutions of quadratic equation vertical motion zero-product property Section 9.5: zero of a function Section 9.6: Trinomial (see Ch 9.1) Section 9.7: perfect square trinomial (see Ch 9.3) Section 9.8: factor completely common factor product of binomial factors Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Monomial 9.1 Meaning DEF: A monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents. Examples Oral Practice The following are monomials: x m2 The following are examples of polynomials: y3, 5mn, 4 and p. ____ and ____ are also examples of monomials. -2xy 5 -2b2c Writing Practice: The following are examples of monomials: 3c, -4f2, 5xy, and 6. The following are examples of monomials: ________________. Degree of Monomial 9.1 The Degree of a Monomial is the sum of the exponents of the variables in the monomial. Monomial Degree x 1 m3 3 2xy 2 -3v12u2 14 5 0 The degree of the monomial 2x3 is 3 because the exponent of the variable x is 3. The degree of the monomial 2xy4 is 5 because the exponent of the variable x is 1 and the exponent of the variable y is 4 and 4 + 1 = 5 The degree of the monomial 12p3k4 is 7 because the exponent of the variable p is 3 and the exponent of the variable k is 4 and 3 + 4 = 7 Writing Practice: The degree of the monomial -x7 is 7 because the exponent of the variable x is 7. The degree of the monomial ______ is _____ because _______________________. Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Polynomial 9.1 Meaning A polynomial is a monomial or a sum of monomials. Examples Oral Practice Below are examples of polynomials 2x3 – 3 is a polynomial. 2x (monomial) -x + 3y – 5 is a polynomial 3x + y (sum of 2 monomials) _______________ is a polynomial, 5m2 – 3m + 2 (sum of 3 monomials) Writing Practice: 3f – 5 is a polynomial. _______ is a polynomial. Degree of Polynomial 9.1 The Degree of a Polynomial is the greatest degree of its terms. Polynomial Degree x+5 1 m3 – 2m2 + 3 3 2xy + 5z5 – 1 5 -3v12 + 6u4v10 + w12 14 2x2 + 3xyz 3 The polynomial 3x4 – 5x3 + 2x3 – 1 is a 4th degree polynomial. The polynomial 2xy – y + 1 is a 2nd degree polynomial. The polynomial ______________ is a ___ degree polynomial. Writing Practice: The polynomial 2x5 – 3x2 – 1 is a 5th degree polynomial. The polynomial __________ is a ___ degree polynomial. Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Leading Coefficient 9.1 Meaning DEF: When a polynomial is written so that the exponents of a variable decrease from left to right, the coefficient of the first term is the leading coefficient. Examples Oral Practice The leading coefficient of -2x5 + 3x3 – 2x + 5 is –2. The leading coefficient of 4v3 – 2v2 + v – 7 is 4. The leading coefficient of 5m4 – 3m2 is 5. The leading coefficient of 2x3 – 5x + 1 is ___. Writing Practice: The leading coefficient of 8m2 – 5m + 1 is 8. The leading coefficient of ______________ is ____. Binomial 9.1 DEF: A polynomial with two terms is called a binomial. The following are examples of binomials. Since the polynomial 2x + 8 has two terms then it is a binomial. 2x + 3 -5mn + 6n Since the polynomial 3m + 5n has _______ then it is a _________. x+y -2x3 + 6xy Writing Practice: Since the polynomial –5v + 7 has two terms then it is a binomial. Since the polynomial _______ has two terms then it is a ________. Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Trinomial 9.1 Meaning A polynomial with three terms is called a trinomial. Examples The following are examples of trinomials Oral Practice Since the polynomial 2x – y + 3 has three terms then it is a trinomial. 2x + 3y – 1 -5mn + 6n + 7 Since the polynomial ________ has ____ terms then it is a _______. x+y+z -2x3 + 6xy – 7 Writing Practice: Since the polynomial 5m + 6n – 1 has three terms then it is a trinomial. Since the polynomial ________ has ____ terms then it is a _______. FOIL Pattern 9.2 This pattern which means F – first, O – outer, I – inner, and L – last helps you remember how to use the distributive property to multiply binomials. To get the product we use FOIL pattern described as: F L For the factors (x – 6)(x + 7), identify the FOIL Products First term products: (x)(x) Outer term products: (x)(7) Inner term products: (-6)(x) Last term products: (-6)(7) (2x + 3) (x + 4) I O First term products: (2x)(x) Outer term products: (2x)(4) Inner term products: (3)(x) Last term products: (3)(4) For the factors (3x + 5)(x – 2), identify the FOIL Products First term products: (3x)(x) Outer term products: (3x)(-2) Inner term products: (5)(x) Last term products: (5)(-2) Writing Practice: For the factors (x + 1)(x + 8), identify the FOIL Products. F – (x)(x) O – (x)(8) I - (1)(x) L – (1)(8) For the factors (_______)(_______) identify the FOIL Products Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Square of a Binomial 9.2 Meaning DEF: Square of a binomial Pattern: (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2 Examples To use the pattern, identify the 1st term a and the 2nd term b. Plug into the formula and simplify: 2 (2x + 3) a = 2x and b = 3 (2x)2+2(2x)(3) + (3)2= 4x2 + 12x + 9 PRODUCT Oral Practice Use the pattern to get the product (x – 5)2 a = x and b = -5 (x)2+2(x)(-5) + (-5)2= x2 – 10x + 25 PRODUCT Use the pattern to get the product (2x – 7)2 a = 2x and b = -7 ( )2+2( )( ) + ( )2= ____________ PRODUCT Writing Practice: Find the product: (3x + 5)2 = 9x2 + 30x + 25 Find the product: (________)2 = _____________ Difference of 2 Squares 9.2 DEF: (a + b)(a – b) = a2 – b2 To use the pattern, identify the 1st term a and the 2nd term b and make sure that they assume the (a + b)(a – b) form. (2x + 3)(2x – 3) = ? a = 2x and b = 3 To use the pattern, identify the 1st term a and the 2nd term b and make sure that they assume the (a + b)(a – b) form. (x – 3)(x + 3) = ? a = x and b = 3 Plug into a and b and simplify: Plug into a and b and simplify: 2 2 2 (2x) – (3) = 4x – 9 (product) (x)2 – (3)2 = x2 – 9 (product) Find the product using the pattern: (2x + 7)(2x – 7) = ( )2 – ( )2 = ________ Writing Practice: Find the product using the pattern: (x + 9)(x – 9) = (x)2 – (9)2 = x2 – 81 (_______)(_______) = ( )2 – ( )2 = ________ Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Perfect Square Trinomial (PST) 9.3 Meaning Examples DEF: a2 + 2ab + b2 = (a + b)2 2 2 a – 2ab + b = (a – b) 2 The following is a perfect square trinomial: 4x2 + 4x + 1 because the middle term is a product of the square root of the first and last term and 2. Square root of 4x2 is 2x and square root of 1 is 1 Oral Practice The trinomial x2 – 6x + 9 is a PST because the middle term is 2(x)(3) and therefore can be factored as (x – 3)2 The trinomial __________ is a PST because the middle term is _____ and therefore can be factored as _________. 2(2x)(1) = 4x, therefore can be factored as (2x + 1)2 Writing Practice: The trinomial 9x2 – 12x + 4 is a PST because the middle term is 2(3x)(2) and therefore can be factored as (3x – 2)2 The trinomial ____________is a PST because the middle term is _______ and therefore can be factored as ________. Roots 9.4 DEF: The solutions of equations. (Such as an equation where one side is zero and the other side is a product of polynomial factors) For the quadratic equation For the quadratic equation (x + 3)(x – 2) = 0 (x + 5)(x + 7) = 0 Since x = -5 or x = -7 Since x = -3 or x = 2 The roots of the equation are –3 and 2 The roots of the equation are –5 and –7 For the quadratic equation (x – 8)(x + 5) = 0 Since x = 8 or x = -5 The roots of the equation are _________. Writing Practice: (x)(x + 2) = 0, Since x = 0 or x = -2. The roots of the equation are 0 and –2. ______________ = 0, Since ______ or ______. The roots of the equation are __________. Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Solutions of Quadratic Equation 9.4 Meaning DEF: The solutions of a quadratic equation the values of the x that satisfies the equation. It varies from none, one or two solutions. Examples Here are examples of Quadratic Equations and their Solutions: x2 – 1 = 0 The solutions of this quadratic equation are x = 1. When these values are plugged into the equation, it makes a true statement. Oral Practice x2 – 2x + 1 = 0 The solution of this quadratic equation is x = 1. When 1 is plugged in the equation it makes the equation true. x2 – 4x + 3 = 0 The solution of this quadratic equation is _____ When ____ is plugged in the equation it makes the equation true. Writing Practice: The solution to x2 – 9 = 0 is 3. When these values are plugged in the equation, it makes the equation true. The solution to ________ is ___. When these values are plugged in the equation, it makes the equation true. The Vertical Motion Model Vertical Motion DEF: An equation for an object thrown upwards with an Model The height h (in feet) of a projectile can initial velocity of 11 ft/sec is given as: 9.4 The Vertical Motion be modeled by: Let v = 10 ft/sec and s = 0 Model can describe the path of a projectile. A h = -16t2 + vt + s h = -16t2 + (11)t + (0) projectile is an object h = -16t2 + 11t thrown into the air but Where t is time in seconds the object has no power to keep has been in the air, v is the initial itself in the air. velocity, and s is the initial height in An equation for an object thrown upwards with an feet. initial velocity of ________ is given as: Let v = _______ and s = 0 The model can be used to find an equation that gives the height of an h = -16t2 + (___)t + (__) object as a function of time (in h = ______________ seconds) given the initial velocity. (s = 0 at this point) Writing Practice: An equation for an object thrown upwards with an initial velocity of 12 ft/sec is given as: h = -16t2 + 12t An equation for an object thrown upwards with an initial velocity of _______ is given as: _____________ Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Meaning Zero-Product Property 9.4 DEF: Let a and b be real numbers. If ab = 0, then a = 0 or b = 0. Examples The zero product property states that: If (x +3)(x – 2) = 0, Oral Practice Fill in the correct conclusion: If (2x – 5)(x + 8) = 0, Then (2x – 5)= 0 or (x + 8) = 0 Then (x +3) = 0 or (x – 2) = 0 Fill in the correct conclusion: If (x – 9)(3x – 7) = 0, Then ______________________ Writing Practice: If (x + 5)(x + 7) = 0, then (x + 5) = 0 or (x + 7) = 0 If ( )( ) = 0, then _____________________ Zero of a Function 9.5 DEF: The zero of a function y = f(x) is a value of x for which f(x) = 0. (or y = 0) Given: Given: F(x) = x – 5 F(x) = 2x – 1 Setting F(x) = 0 and solving for x, the value of x=5 Therefore, 5 is a zero of the function because F(5) = 0 Setting F(x) = 0 and solving for x, the value of x = 1/2 Therefore, 1/2 is a zero of the function because F(1/2) = 0 F(x) = 2x + 6 Setting F(x) = 0 and solving for x, the value of x = -3 Therefore, ____________________ because ________ Writing Practice: If F(-2) = 0, then –2 is a zero of the function. If _______, then ___ is a zero of the function. Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Factor Completely 9.8 Meaning DEF: A factorable polynomial with integer coefficients is said to be factored completely if it is written as a product of unfactorable polynomials with integer coefficients. Examples Given 2x2 – 5x – 12 = (2x + 3)(x – 4) Since it is written as (2x + 3)(x – 4) with integer coefficients, then the polynomial 2x2 – 5x – 12 is factored completely. Oral Practice Given 15x2 + 11x + 2 = (3x + 1)(5x + 2) Since it is written as (3x + 1)(5x + 2) with integer coefficients, then the polynomial 15x2 + 11x + 2 is factored completely. Given 5x2 – 14x – 3 = (x – 3)(5x + 1) Since it is written as (x – 3)(5x + 1) with integer coefficients, then the polynomial 5x2 – 14x – 3 is factored completely. Writing Practice: Given 2x2 + 5x – 3 = (x + 3)(2x – 1). Since the factors have integer coefficients, then it is factored completely. Given __________ = ____________. Since the factors have _____________, then ___________________. Common Factor 9.8 DEF: A common factor is a whole number that is a factor of each number. Given the numbers and their factors: 25: 1, 5, 25 10: 1, 2, 5, 10 The number 5 is a common factor. Given the numbers and their factors: 12: 1, 2, 3, 4, 6, 12 20: 1, 2, 4, 5, 10, 20 The numbers 3 and 4 are common factors. Given the numbers and their factors: 24: 1, 2, 3, 4, 6, 8, 12, 24 36: 1, 2, 3, 4, 6, 9, 12, 36 The numbers _____________ are common factors. Writing Practice: The common factor(s)of 12 and 36 are 2, 3, 4, 6, and 12 The common factor(s)of ____ and ____ are _____________ Permission for Use Granted by Dr. Kate Kinsella 10/08 Word Product of Binomial Factors 9.8 Meaning DEF: If a polynomial is factored completely and the factors are written as a binomial, the factors are said to be written as product of binomial factors. Examples The factors of 15x2 + 11x + 2 are written as product of binomial factors as (3x + 1)(5x + 2) because the factors are binomials. Oral Practice The factors of 5x2 – 14x – 3 are written as product of binomial factors as (x – 3)(5x + 1) because the factors are binomials. The factors of ___________are written as product of binomial factors as ____________ because the factors are binomials. Writing Practice: The factors of 2x2 + 5x – 3 are written as product of binomial factors as (x + 3)(2x – 1) because the factors are binomials. The factors of __________ are written as product of binomial factors as ____________ because the factors are binomials. Permission for Use Granted by Dr. Kate Kinsella 10/08