Lecture 281 Unit 4 Lecture 28 Introduction to Quadratics Introduction to Quadratics

Lecture 282 Objectives Evaluate quadratic expressionsEvaluate quadratic expressions Identify the degree of a polynomialIdentify the degree of a polynomial Determine the number of terms in a polynomialDetermine the number of terms in a polynomial Add and subtract polynomialsAdd and subtract polynomials Multiply binomials using FOILMultiply binomials using FOIL

Lecture 283 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h = -16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t

Lecture 284 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t 2 -16(2) (2) =

Lecture 285 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t 2 -16(2) (2) = =

Lecture 286 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t 2 -16(2) (2) = = (5) (5) = 8 10

Lecture 287 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t 2 -16(2) (2) = = (5) (5) = =

Lecture 288 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t 2 -16(2) (2) = = (5) (5) = = (8) (8) = 10

Lecture 289 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t 2 -16(2) (2) = = (5) (5) = = (8) (8) = =

Lecture 2810 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t 2 -16(2) (2) = = (5) (5) = = (8) (8) = = (10) (10) =

Lecture 2811 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t 2 -16(2) (2) = = (5) (5) = = (8) (8) = = (10) (10) = = 0

Lecture 2812 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. When will the arrow reach its maximum height? th

Lecture 2813 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. When will the arrow reach its maximum height? th When time = 5 sec, height = 400 feet.

Lecture 2814 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. When will the arrow hit the ground? th

Lecture 2815 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h=-16t t t is the number of seconds the arrow is in the air. When will the arrow hit the ground? th When time = 10 sec, height = 0 feet.

Lecture 2816 h = -16t t h = -16t t Timeth Height

Lecture 2817 Term: Definitions

Lecture 2818 Term: A number or the product of a number and a variable raised to a power. Definitions

Lecture 2819 Term: A number or the product of a number and a variable raised to a power. Example: 2x 3, 3, 4x, 5x 6, y, x Definitions

Lecture 2820 Term: A number or the product of a number and a variable raised to a power. Example: 2x 3, 3, 4x, 5x 6, y, x Like Terms Definitions

Lecture 2821 Term: A number or the product of a number and a variable raised to a power. Example: 2x 3, 3, 4x, 5x 6, y, x Like Terms Terms with the same variables raised to exactly the same powers. Definitions

Lecture 2822 Term: A number or the product of a number and a variable raised to a power. Example: 2x 3, 3, 4x, 5x 6, y, x Like Terms Terms with the same variables raised to exactly the same powers. Example: 3x 2, 4x 2 Definitions

Lecture 2823 How do we combine like terms?

Lecture 2824 How do we combine like terms? Add the numerical coefficients and multiply the result by the common variable factor.

Lecture 2825 How do we combine like terms? Add the numerical coefficients and multiply the result by the common variable factor. -8x 2 + 2x – 8 – 6x x + 2

Lecture 2826 How do we combine like terms? Add the numerical coefficients and multiply the result by the common variable factor. -8x 2 + 2x – 8 – 6x x + 2

Lecture 2827 How do we combine like terms? Add the numerical coefficients and multiply the result by the common variable factor. -8x 2 + 2x – 8 – 6x x x x – 6

Lecture 2828 Simplify: 2(3x 2 + 5x – 10) – 4(x 2 – 6x + 3)

Lecture 2829 Simplify: 2(3x 2 + 5x – 10) – 4(x 2 – 6x + 3) 6x x – 20 – 4x x - 12

Lecture 2830 Simplify: 2(3x 2 + 5x – 10) – 4(x 2 – 6x + 3) 6x x – 20 – 4x x – 12 6x x – 20 – 4x x - 12

Lecture 2831 Simplify: 2(3x 2 + 5x – 10) – 4(x 2 – 6x + 3) 6x x – 20 – 4x x – 12 2x x – 32 6x x – 20 – 4x x - 12

Lecture 2832 Simplify: 5(2x 2 + 5) – (8x 2 + 2x – 4)

Lecture 2833 Simplify: 5(2x 2 + 5) – (8x 2 + 2x – 4) 10x – 8x 2 – 2x + 4

Lecture 2834 Simplify: 5(2x 2 + 5) – (8x 2 + 2x – 4) 10x – 8x 2 – 2x + 4

Lecture 2835 Simplify: 5(2x 2 + 5) – (8x 2 + 2x – 4) 2x 2 – 2x x – 8x 2 – 2x + 4

Lecture 2836 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x x Cost: C = 3x x + 40

Lecture 2837 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x x P = (Revenue) – (Cost) Cost: C = 3x x + 40

Lecture 2838 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x x P = (-5x x) – (Cost) Cost: C = 3x x + 40

Lecture 2839 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x x P = (-5x x) – (3x x + 40) Cost: C = 3x x + 40

Lecture 2840 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x x P = (-5x x) – (3x x + 40) Cost: C = 3x x + 40 P = – 5x x – 3x x – 40

Lecture 2841 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x x P = -8x 2 P = (-5x x) – (3x x + 40) Cost: C = 3x x + 40 P = – 5x x – 3x x – 40

Lecture 2842 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x x P = - 8x x P = (-5x x) – (3x x + 40) Cost: C = 3x x + 40 P = – 5x x – 3x x – 40

Lecture 2843 The equation for profit is, Profit = Revenue - Cost Revenue: R = -5x x P = - 8x x - 40 P = (-5x x) – (3x x + 40) Cost: C = 3x x + 40 P = – 5x x – 3x x – 40

Lecture 2844 Polynomial: Definitions

Lecture 2845 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Definitions

Lecture 2846 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Examples: 2x 3 Definitions

Lecture 2847 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Examples: 2x 3 3 +x Definitions

Lecture 2848 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Examples: 2x 3 3 +x 7+x+5x 6 Definitions

Lecture 2849 Polynomial: A term or a finite sum of terms in which variables may appear in the numerator raised to whole number powers only. Examples: 2x 3 3 +x 7+x+5x 6 x Definitions

Lecture 2850 Polynomials:Nonpolynomials: Definitions

Lecture 2851 Polynomials:Nonpolynomials: Definitions

Lecture 2852 Polynomials:Nonpolynomials: Definitions

Lecture 2853 Polynomials:Nonpolynomials: Definitions

Lecture 2854 Polynomials:Nonpolynomials: Definitions

Lecture 2855 Polynomials:Nonpolynomials: Definitions

Lecture 2856 Polynomials:Nonpolynomials: Definitions

Lecture 2857 Polynomials:Nonpolynomials: Definitions

Lecture 2858 Monomial: Examples: Binomial: Examples: Trinomial: Examples: Definitions

Lecture 2859 Monomial: A polynomial with exactly 1 term. Examples: Binomial: Examples: Trinomial: Examples: Definitions

Lecture 2860 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: Examples: Trinomial: Examples: Definitions

Lecture 2861 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: A polynomial with exactly 2 terms. Examples: Trinomial: Examples: Definitions

Lecture 2862 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: A polynomial with exactly 2 terms. Examples: 2x 3 +9x, 3 +x, 7+5x 6 Trinomial: Examples: Definitions

Lecture 2863 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: A polynomial with exactly 2 terms. Examples: 2x 3 +9x, 3 +x, 7+5x 6 Trinomial: A polynomial with exactly 3 terms. Examples: Definitions

Lecture 2864 Monomial: A polynomial with exactly 1 term. Examples: 2x 3, x, 7, y, 5x 6 Binomial: A polynomial with exactly 2 terms. Examples: 2x 3 +9x, 3 +x, 7+5x 6 Trinomial: A polynomial with exactly 3 terms. Examples: 3x 2 +6x+2, 7+5x 3 +4x 2 Definitions

Lecture 2865 Degree of a term: Examples:Definitions

Lecture 2866 Degree of a term: The sum of the exponents on the variables in the term. Examples:Definitions

Lecture 2867 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3

Lecture 2868 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3

Lecture 2869 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6

Lecture 2870 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6

Lecture 2871 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6 x

Lecture 2872 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6 x1

Lecture 2873 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6 x1 xy

Lecture 2874 Degree of a term: The sum of the exponents on the variables in the term. Examples:DefinitionsTermDegree 2x 3 3 5x 6 6 x1 xy2

Lecture 2875 Degree of a polynomial: Examples:Definitions

Lecture 2876 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:Definitions

Lecture 2877 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9

Lecture 2878 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3

Lecture 2879 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7

Lecture 2880 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6

Lecture 2881 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6 x + y + 6

Lecture 2882 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6 x + y + 6 1

Lecture 2883 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6 x + y xy + 5x - 9y

Lecture 2884 Degree of a polynomial: Greatest degree of any term of the polynomial Examples:DefinitionsPolynomialDegree 2x 3 +7x+9 3 5x 6 –2x 4 +9x -7 6 x + y xy + 5x - 9y 2

Lecture 2885 What is a binomial?

Lecture 2886 What is a binomial? A polynomial with exactly two terms.

Lecture 2887 What is a binomial? A polynomial with exactly two terms. Examples: x + y, 3 + x, 4x 2 + 9

Lecture 2888 To Multiply Binomials, use FOIL: (ax+b)(cx+d)

Lecture 2889 To Multiply Binomials, use FOIL: acx 2 + (ax+b)(cx+d) F F

Lecture 2890 To Multiply Binomials, use FOIL: acx 2 + adx (ax+b)(cx+d) F O O F

Lecture 2891 To Multiply Binomials, use FOIL: acx 2 + adx + bcx (ax+b)(cx+d) F O I I O F

Lecture 2892 To Multiply Binomials, use FOIL: acx 2 + adx + bcx + bd (ax+b)(cx+d) L F O I L I O F

Lecture 2893 To Multiply Binomials, use FOIL: acx 2 + adx + bcx + bd (ax+b)(cx+d) acx 2 + (ad+bc)x + bd L F O I L I O F

Lecture 2894 Multiply: (x+5)(x+3) (x+5)(x+3)

Lecture 2895 Multiply: x2x2x2x2 (x+5)(x+3) F

Lecture 2896 Multiply: x 2 + 3x (x+5)(x+3) F O

Lecture 2897 Multiply: x 2 + 3x + 5x (x+5)(x+3) F O I

Lecture 2898 Multiply: x 2 + 3x + 5x + 15 (x+5)(x+3) F O I L

Lecture 2899 Multiply: x 2 + 3x + 5x + 15 (x+5)(x+3) x 2 + 8x + 15 F O I L

Lecture Multiply: (x-6)(x+2)

Lecture Multiply: (x-6)(x+2) x2x2x2x2 (x-6)(x+2)

Lecture Multiply: (x-6)(x+2) x 2 + 2x (x-6)(x+2)

Lecture Multiply: (x-6)(x+2) x 2 + 2x - 6x (x-6)(x+2)

Lecture Multiply: (x-6)(x+2) x 2 + 2x - 6x - 12 (x-6)(x+2)

Lecture Multiply: (x-6)(x+2) x 2 + 2x - 6x - 12 (x-6)(x+2) x 2 - 4x - 12

Lecture Multiply: (x-7)(x-5)

Lecture Multiply: (x-7)(x-5) x2x2x2x2 (x-7)(x-5)

Lecture Multiply: (x-7)(x-5) x 2 - 5x (x-7)(x-5)

Lecture Multiply: (x-7)(x-5) x 2 - 5x - 7x (x-7)(x-5)

Lecture Multiply: (x-7)(x-5) x 2 - 5x - 7x + 35 (x-7)(x-5)

Lecture Multiply: (x-7)(x-5) x 2 - 5x - 7x + 35 (x-7)(x-5) x x + 35

Lecture Multiply: 6x 2 (2x+5)(3x-8)

Lecture Multiply: 6x x (2x+5)(3x-8)

Lecture Multiply: 6x x + 15x (2x+5)(3x-8)

Lecture Multiply: 6x x + 15x - 40 (2x+5)(3x-8)

Lecture Multiply: 6x x + 15x - 40 (2x+5)(3x-8) 6x 2 - x - 40

Lecture Multiply: 6x x + 15x - 40 (2x+5)(3x-8) 6x 2 - x - 40

Lecture Multiply: (3x+4) 2 9x 2 (3x+4)(3x+4)

Lecture Multiply: (3x+4) 2 9x x (3x+4)(3x+4)

Lecture Multiply: (3x+4) 2 9x x + 12x (3x+4)(3x+4)

Lecture Multiply: (3x+4) 2 9x x + 12x + 16 (3x+4)(3x+4)

Lecture Multiply: (3x+4) 2 9x x + 12x + 16 (3x+4)(3x+4) 9x x + 16

Lecture 28123

Lecture 28124

Lecture 28125