Lesson-4-Algebra.pptx

MMW-ALGEBRA SEPTEMBER 25,2024  DIVISION OF POLYNOMIALS Dividing polynomials is an 1 algorithm to solve a rational number 2 that represents a polynomial divided by a monomial 3 or another polynomial. The divisor and the dividend are placed exactly the same way as we do for regular divisi...

MMW-ALGEBRA SEPTEMBER 25,2024  DIVISION OF POLYNOMIALS Dividing polynomials is an 1 algorithm to solve a rational number 2 that represents a polynomial divided by a monomial 3 or another polynomial. The divisor and the dividend are placed exactly the same way as we do for regular division. BY LONG DIVISION: Step 1. Divide the first term of the dividend () by the first term of the divisor (x), and put that as the first 1 term in the quotient (4x). Step 2 2. Multiply the divisor by that answer, place the product ( - 12x) below the dividend. 3 3. Subtract to create a new polynomial (7x - 21). Step Step 4. Repeat the same process with the new polynomial obtained after subtraction. Problem 1 #1: a. 2 b. c. 3 BY SYNTHETIC DIVISION: 1 Synthetic division is a technique to divide 2 a polynomial with a linear binomial by only considering the values of the coefficients. 3 Step 1: Write the divisor in the form of x - k and write k on the left side of the division. Here, the divisor is x-4, so1 the value of k is 4. Step 2: Set up the division by writing the coefficients of 2 dividend on the right and k on the left. [Note: Use the 0's for the missing terms in the dividend] 3 Step 3: Now, bring down the coefficient of the highest degree term of the dividend as it is. Here, the leading coefficient is 1 (coefficient of ). Step 4: Multiply k with that leading coefficient and write the product below the second coefficient from the left side of the dividend. So, we get, 4×1=4 that we will write below 0. 1 5: Add the numbers written in the second column. Here, by adding Step we get 0+4=4. 2 6: Repeat the same process of multiplication of k with the number Step obtained in step 5 and write the product in the next column to the right. 3 7: At last, we will write the final answer which will be one degree less Step than the dividend. So, here, in our dividend, the highest degree term is x2, therefore, in the quotient, the highest degree term will be x. Therefore, the answer obtained is x+4+(19/x-4). 1 Problem #2: a. 2 b. 3 c. WHAT IS QUADRATIC EQUATION? A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its 1 standard form is a + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the 2 constant term. The important condition for an equation to be a quadratic equation is the coefficient of 3is a non-zero term (a ≠ 0). For writing a quadratic equation in standard form, the term is written first, followed by the x term, and finally, the constant term is written. 1 2 3 Roots of a Quadratic Equation 1 The roots of a quadratic equation are the two values of x, which are obtained by solving the 2 quadratic equation. These roots of the quadratic equation are also called the zeros of 3 equation. the Roots 1 can be obtained by the following: a.By 2 factoring b.By 3 quadratic formula c. Completing the square Nature of Roots of the Quadratic Equation The nature of roots of a quadratic equation can be1 found without actually finding the roots (α, β) of the equation. This is possible by taking the 2 discriminant value, which is part of the formula to solve the quadratic equation. The value b2 - 4ac is called the discriminant of a 3 quadratic equation and is designated as 'D'. Based on the discriminant value the nature of the roots of the quadratic equation can be predicted. Discriminant: D = b - 4ac 2 D > 0, the roots are real and 1 distinct 2 D = 0, the roots are real and equal. 3 D < 0, the roots do not exist or the roots are imaginary. Sum and Product of Roots of Quadratic 1 Equation Sum of the Roots: -b/a = - 2 Coefficient of x/ Coefficient of x 2 Product 3 of the Roots: = c/a = Constant term/ Coefficient of x 2 The 1 quadratic equation can also be formed for the given roots of the equation. If α, β, are 2 the roots of the quadratic equation, then the quadratic equation is as follows. x 3 - (α + β)x + αβ = 0 2 Problem #3: 1 a. What is the quadratic equation 2 whose roots are 4 and -1. 3 Problem #4: Solve 1 the roots of the following using 2 factoring: a. b. 3 Problem #5: Solve 1 the roots of the following using 2 quadratic formula: a. b. 3 Problem #6: Solve 1 the roots of the following using completing the square: 2 a. b. 3 c. 1 2 3 1 THANK YOU 2 3 FOR LISTENING!!!