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BMC MTH101 Ms.Afaf Alqahtani Lecture 4 Outlines: Variables and Constants Algebraic Expressions and Polynomials Coefficient Polynomials Variables and Constants Example of constant: Example of variables: Variable Constance Variables and Constants Variable...

BMC MTH101 Ms.Afaf Alqahtani Lecture 4 Outlines: Variables and Constants Algebraic Expressions and Polynomials Coefficient Polynomials Variables and Constants Example of constant: Example of variables: Variable Constance Variables and Constants Variables Constants Example of constant: Example of variables: Variable Constance Variables and Constants. (+ , - , x, —. ) Variable Constance Variables and Constants Variables, Constants , Coefficient Operations Elgebraic expression 3 Polynomials No variables O Polynomials Fractions or Negative numbers or variable under root W={ 0, 1, 2, ……} , , not real numbers X, y , Z Polynomials Variables and Constants Exercise: o Which of the following is the number of. terms in these expressions: (+ , - , x, —. ) 1) 2xy+3x 0 , 1 , 2 , 3 1) 2x2 y 3 +x3 y – x + 9 1 , 2 , 3 , 4 o What is the coefficient of x in -9x o Variable Constance What is the coefficient of y in y o Which of the following are constant 3x , -4 , , 12z Variables and Constants Exercise: o Which of the following are polynomials. (+ , - , x, —. ) o Which of the following are not polynomials o Which of the following is the number of variables in these polynomial: 1)Variable 2x2+3x Constance 0 , 1 , 2 , 3 2) 2x2 y 3 +x3 y – x + 9 0 , 1 , 2 , 3 Thank You BMC MTH101 Ms.Afaf Alqahtani Lecture 5 Outlines: Degree of terms and polynomials Classifying Polynomials Evaluation of a Polynomial Zero of polynomial  Degree of term and polynomials * A polynomial may have many terms. Every term has a degree. The degree of the term The degree of the term is the sum of the exponents of the variables in the term.  Degree of term and polynomials  Degree of term and polynomials  Degree of term and polynomials Exercise: 1) If 2𝑥4𝑦3− 9𝑥3𝑦2 + 6𝑥𝑦 + 8 is polynomial, then the degree of this term -9x3y2 is: o 2 o 3 o 5 o 7 2) Which of the following is the degree of the polynomial 2𝑥4𝑦3− 9𝑥3𝑦2 + 6𝑥𝑦 + 8 : o 2 o 3 o 5 o 7 3) Which of the following is the degree of the constant 8 in the polynomial 2𝑥4𝑦2 + 6𝑥𝑦 + 8 : o 0 o 3 o 4 o 7 Classification of Polynomials 1) Classification of Polynomials ( By degree ) Classifying Polynomials 2) Classification of Polynomials ( by number of terms ): Summary Classification of Polynomials By number of terms By degree Polynomials Exercise What is the name of the following polynomials: Polynomials Classifying Polynomials Exercise What are the following polynomials called : 1) 2) 3) Classifying Polynomials Exercise Which of the following are monomials? Which of the following are binomials? Which of the following are Trinomials? Evaluation of a Polynomial Evaluation of a Polynomial Exercise Which of the following is the value of this polynomial P(x) = at x = -1 o P(-1) = 0 o P(-1) = 3 o P(-1) = -3 o P(-1) = -6 Evaluation of a Polynomial Exercise Which of the following is the value of this polynomial P(x) = at x = 0 o P(0) = 0 o P(0) = 3 o P(0) = -3 o P(0) = -6 Evaluation of a Polynomial (Zero of a Polynomial) Example Which of the following is the value of this polynomial P(x) = , at x = 0 o P(0) = 0 o P(0) = 2 o P(0) = -3 o P(0) = -6 Which of the following is the value of this polynomial P(x) = , at x = 1 o P(1) = 0 o P(1) = 3 o P(1) = -3 o P(1) = -6 Which of the following is the value of this polynomial P(x) = , at x = 2 o P(2) = 0 o P(2) = 3 o P(2) = -3 o P(2) = -6 Evaluation of a Polynomial (Zero of a Polynomial) Note: P(0) = 2 ≠ 0 ➔ x = 0 is not zero of the polynomial P(x)= Evaluation of a Polynomial (Zero of a Polynomial) X= X=-2-2 Evaluation of a Polynomial (Zero of a Polynomial) Exercise 1) Which of the following is the zero of this polynomial P(x) = 3 x - 12 o x=0 o x=4 o x = -3 o x = -6 2) If P(x) = x3 – 6 x + 11 x – 6 , then the zero of this polynomial is: o x=0 o x=1 o x = -3 o x = -6 Thank You BMC MTH101 Ms.Afaf Alqahtani Lecture 6 Outlines: Linear Equations in one variable Consecutive Integers Linear Equations in one variable Example (1): Solve the following equation: 2 x - 3 = 13 Solution: 2x=13+3 2x=16 x=8 ➔What is the principle we used to solve the above linear equation in one variable? The first technique for solving linear equation in one variable is Addition/Subtraction Rule. which states: If c is any real is any real number, the following equations are equivalent (they have the same solution set) a=b a±c=b±c The second line states that, adding or subtracting the same number to both side of an equation will not change the solution set of the equation. Linear Equations in one variable Example (1) :Which of the following is the correct answer: If a, b and c are real numbers, and a=b then a+b=b+c a+c=b+c a+c=a–b a–b=b–c Example (2) : Which of the following is the solution of this equations: a) x – 5 = 12 9 10 17 7 b) x + 25 = 12 – 13 37 –5 10 Linear Equations in one variable Multiplication-Division Rule: The second technique for solving linear equation in one variable is the Multiplication- Division rule. which states: If c is any real is any real number, the following equations are equivalent (they have the same solution set) a=b a×c=b×c 𝒂 𝒃 = ,𝐜 ≠ 𝟎 𝒄 𝒄 If we multiply or divide both sides of an equation by any real number, does not effect the solution of the equation. Linear Equations in one variable Example (3) :Which of the following is the correct answer: If a, b and c are real numbers, and a=b then a×b=b×c a×c=b×c 𝒂 𝒃 = 𝒄 𝒂 𝒂 𝒃 = c𝒂 𝒃 Example (4) : Which of the following is the solution of this equations: 𝒙 =–4 𝟓 3 –20 10 7 Linear Equations in one variable Example (5) : Which of the following is the solution of this equations: 10x+9=3x – 5 –2 3 5 –10 Linear Equations in one variable Example (6) : Which of the following is the solution of this equations: 2x+7=19 2 3 6 –1 Consecutive Integers Consecutive integers are integers that Follow each other in order. For example, 1, 2, 3, and 4 are consecutive integers. We deal specifically with integers that are: consecutive integers consecutive odd integers consecutive even integers. The condition of being consecutive allows us to get the necessary relationships among the variables to be able to get one equation in one unknown.  CONSECUTIVE CONSECUTIVE CONSECUTIVE INTEGERS EVEN INTEGERS ODD INTEGERS the difference the difference the difference is 1 is 2 is 2 n = first n = first n = first Let consecutive Let consecutive even integer Let consecutive odd integer integer n+1 =second n+2 = second n+2 = second consecutive consecutive consecutive integer even integer odd integer n + 2 = third n+4 = third n+4 = third consecutive consecutive consecutive integer even integer odd integer and so on. and so on. and so on. Consecutive Integers The difference is one: Examples: 1, 2, 3, 4, ……. 200, 201, 202, 203, …… Consecutive Even Integers The difference is two: 22 4 6 8 Examples: 2, 4, 6, 8, ……. 12, 14, 16, 18, …. Consecutive Odd Integers The difference is two: 22 4 6 8 Examples: 3,5,7,9 ……. 13, 15, 17, 19…. Consecutive Integers Example(7): What are the four consecutive integers whose sum is 66. 15, 16,17 and 18 3, 4, 5 and 6 5,4, 6 and 7 –10, –11, –12 and –13 Consecutive Integers Example(8): What is the three consecutive even integers whose sum is 162. 25, 30 and 35 52, 54 and 56 55, 60 and 70 20, 24 and 26 Consecutive Integers Example(9): What is the three consecutive odd integers whose sum is 159. 51, 53 and 55 52, 54 and 56 55, 60 and 65 20, 24 and 26 Exercises -Which of the following are the four consecutive odd integers whose sum is 152. 35, 37, 39 and 41 52, 54 ,56 and 58 55, 57, 59 and 61 20, 22, 24 and 26 -Which of the following are the three consecutive integers whose sum is 567. 188, 189 and 190 520, 540 and 560 218, 219, and 220 200, 201 and 202 Exercises -Which of the following is the difference in consecutive odd integers : 0 1 2 3 -Which of the following is the difference in consecutive integers : 0 1 2 3 Exercises -Which of the following are the three consecutive odd integers have a sum of 57. 5, 7 and 9 7, 9 and 11 15, 16 and 17 17,19 and 21 -Which of the following are the sum of four consecutive integers is 406. 51, 58, 65 and 71 91, 98, 105 and 112 7, 14, 21 and 28 100,101,102 and 103 Exercises -Which of the following is the solution of 2)𝑥−4) +6=4(𝑥+5) 10 11 6 –11 3𝑥+8=2 –2 3 7 1 2 (3𝑥+4) =14−3(𝑥−1) 1 2 3 1 12 Thank You BMC MTH101 Ms.Afaf Alqahtani Lecture 7 Outlines -Quadratic Equations In one Variable Quadratic Equations The general form of a quadratic equation in one variable is: Where a, b and c are real numbers a is NOT zero because if 𝑎= 0, then the equation is no more quadratic. If a=0 (0) 𝑥 2 +b𝑥 + 𝑐 = 0 0+b𝑥 + 𝑐 = 0 b𝑥+c=0 (linear equation) Note: Quadratic equations will have 0, 1 or 2 real solutions Quadratic Equations Example: -In the quadratic equation in one variable 3x2 +4x -5=0 ; we have a = 3 , b = 4 , c = -5 Exercise (1): Which of the following are the coefficients a, b and c in the quadratic equation x2 - 3x = -2 Solution: the general form of a quadratic equation x2 - 3x = - 2 is x2 - 3x + 2 = 0 Then, a=1 , b= -3, c= 2 Exercise (2): Which of the following are the coefficients a, b and c in the quadratic equation 5x2 + 3x -7 = 0 Solution: a = 5 , b = 3 , c = -7 Quadratic Equations We use the following formula to find the solution/s of a quadratic equation in one variable. The quadratic formula 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 Where a is a coefficient of 𝑥 2 b is a coefficient of 𝑥 c is a constant Solution is Exercises Equations Quadratic Example (1): Which of the following are the solutions of the quadratic equation 2𝑥 2 − 3𝑥 + 1 = 0 x=1 or x=5 1 x=1 or x= 2 x=7 or x=2 3 x= or x= 3 2 Example (2): Which of the following is the number of solutions of the quadratic equation 2𝑥 2 − 3𝑥 + 1 = 0 1 solution 2 solutions 3 solutions No solution Exercises Quadratic Equations Example (2): Which of the following is the number of solutions of the quadratic equation 2𝑥 2 − 3𝑥 + 1 = 0 1 solution 2 solutions 3 solutions No solution Exercises Equations Quadratic Example (1): Which of the following are the solutions of the quadratic equation 2𝑥 2 − 3𝑥 + 1 = 0 x=1 or x=5 Solution steps: a=2, b= −3, c=1 1 x=1 or x= 2 Δ = 𝑏 2 − 4𝑎𝑐 x=7 or x=2 = (−3 )2 −4(2)(1) 3 x= 2 or x= 3 =9−8 = 1 >0 two solutions −𝒃 − 𝚫 −𝒃 + 𝚫 𝒙𝟏 = , 𝒐𝒓 𝒙𝟐 = 𝟐𝒂 𝟐𝒂 −(−𝟑) − 𝟏 −(−𝟑) + 𝟏 𝒙𝟏 = , 𝐨𝐫 𝒙𝟐 = 𝟐(𝟐) 𝟐(𝟐) Example (2): Which of the following is the number of solutions of the quadratic equation 2𝑥 2 − 3𝑥 + 1 = 0 1 solution 2 solutions 3 solutions No solution Exercises Equations Quadratic Example (3): Which of the following is the solution of 𝒙𝟐 + 𝟐𝒙 + 𝟏 = 𝟎 x= -1 No real solution x= 6 x= 4 Example (4): Which of the following is the number of solutions of the quadratic equation 𝒙𝟐 + 𝟐𝒙 + 𝟏 = 𝟎 1 solution 2 solutions 3 solutions No solution Exercises Quadratic Equations Example (4): Which of the following is the number of solutions of the quadratic equation 𝒙𝟐 + 𝟐𝒙 + 𝟏 = 𝟎 1 solution 2 solutions 3 solutions No solution Exercises Equations Quadratic Example (3): Which of the following is the solution of 𝑥 2 + 2𝑥 + 1 = 0 x= -1 Solution steps: a=1, b= 2, c=1 No real solution Δ = 𝑏 2 − 4𝑎𝑐 = (2)2 −4(1)(1) x= 6 = 4 −4 =0 one solution x= 4 𝑥 = −𝒃 = −𝟐 = −1 𝟐𝒂 𝟐(𝟏) Example (4): Which of the following is the number of solutions of the quadratic equation 2𝑥 2 − 3𝑥 + 1 = 0 1 solution 2 solutions 3 solutions No solution Exercises Equations Quadratic Example (5): Which of the following is the solution of 2𝑥 2 − 2𝑥 + 5 = 0 x= 11 No real solution x= 7 x= 3 Example (6): Which of the following is the number of solutions of the quadratic equation 2𝑥 2 − 2𝑥 + 5 = 0 1 solution 2 solutions 3 solutions No solution Exercises Quadratic Equations Example (6): Which of the following is the number of solutions of the quadratic equation 2𝑥 2 − 2𝑥 + 5 = 0 1 solution 2 solutions 3 solutions No solution Exercises Equations Quadratic Example (5): Which of the following is the solution of 2𝑥 2 − 2𝑥 + 5 = 0 x= 11 Solution steps: a=2, b= −2, c=5 No real solution Δ = 𝑏 2 − 4𝑎𝑐 x= 7 = (−2 )2 −4(2)(5) x= 3 =4−40 = − 36 < 0 the quadratic equation has no real solution. Example (6): Which of the following is the number of solutions of the quadratic equation 2𝑥 2 − 2𝑥 + 5 = 0 1 solution 2 solutions 3 solutions No solution Exercises Exercise(1): Which of the following are the solutions of the quadratic equation 𝑥 2 − 4𝑥 + 3 = 0 x=1 or x=3 x=2 or x=3 x=6 or x=2 x=6 or x=1 Exercise (2): Which of the following is the number of solutions of the quadratic equation 𝑥 2 − 4𝑥 + 3 = 0 1 solution 2 solutions 3 solutions No solution Exercises Exercise(3): Which of the following are the solutions of the quadratic equation 5x2 + 4x + 4 =0 x= 11 1 or 3 No real solution 2 or 3 6x=or 27 6x= or 13 x=1 or x=3 x=2 or x=3 x=6 or x=2 x=6 or x=1 Exercise (4): Which of the following is the number of solutions of the quadratic equation 5x2 + 4x + 4 =0 1 solution 2 solutions 3 solutions No solution Exercises Exercise(5): Which of the following are the solutions of the quadratic equation 2 2𝑥 x + 6x− +9 4𝑥 =0 +3 =0 x= 11 Solution steps: 𝑥 2 − 4𝑥 + 3 = 0 1 or 3 No real solution a=1, b= −4, c=3 2 or 3 6x=or 27 Δ = 𝑏 2 − 4𝑎𝑐 = (−4 )2 −4(1)(3) 6x= or 1- 3 =16 −12 = 4 >0 two solutions −𝒃 − 𝚫 −𝒃 + 𝚫 𝒙𝟏 = , 𝒐𝒓 𝒙𝟐 = 𝟐𝒂 𝟐𝒂 −(−𝟒) − 𝟒 −(−𝟒) + 𝟒 𝒙𝟏 = , 𝐨𝐫 𝒙𝟐 = 𝟐(𝟏) of solutions of the quadratic Exercise (6): Which of the following is the number 𝟐(𝟏) equation 𝟒−𝟐 𝟒+𝟐 = x2 + 6x = +9 𝟏 =0 𝐨𝐫 = =3 𝟐 𝟐 1 solution 2 solutions 3 solutions No solution Thank You BMC MTH101 Ms.Afaf Alqahtani - Self Evaluation CH 2 Variables, Constants , Coefficient Variables, Constants , Coefficient Thank You