POLYNOMIALS and different types of polynomial.pptx

polynomial

1. POLYNOMIALS

2. POLYNOMIALS a. POLYNOMIAL functions and their graphs b. dividing polynomials c. Real zeros of polynomials

3. WHAT IS POLYNOMIAL Polynomial is made up of two words, poly and nomial . “ Poly ” means many and “ nomial ” means the “ term ”, and hence when they are combined, we can say that polynomials are “algebraic expressions with many terms”.

4. POLYNOMIAL functions Are expressions that may contain variables of varying degrees, coefficients , positive exponents , and constants . f(x) = 3x 2 - 5 Examples: g(x) = -7x 3 + ½ x - 7 h(x) = 3x 4 + 7x 3 - 12x 2

5. POLYNOMIAL functions in standard form f(x) = a n x n + a n-1 x n-1 + … + a 2 x 2 + a 1 x + a . a n can’t be equal to zero and is called the leading coefficient a n ,a n-1 , … a are real number constants This algebraic expression is called a polynomial function in variable x. n is a non-negative integer Each exponent of variable in polynomial function should be a whole number.

6. How to determine POLYNOMIAL functions Polynomial (degree 3) - cubic Polynomial ( degree 7) Polynomial (degree 2) - quadratic Not a polynomial Not a polynomial f(x) = 4x 3 – 3x 2 + 2 f(x) = x 7 – 4x 5 + 1 f(x) = 4x 2 – 2x – 4 f(x) = 4x 3 + √x – 1 f(x) = 5x 4 – 2x 2 + 3/x Note: Remember that coefficients can be fractions, negative numbers, 0, or positive numbers. We just need to take care of the exponents of variables to determine whether it is a polynomial functions.

7. How to determine POLYNOMIAL functions The exponent of the variable in the function in every term must only be a non-negative whole number. i .e., the exponent of the variable should not be a fraction or negative number. The variable of the function should not be inside a radical i .e., it should not contain any square roots, cube roots, etc. The variable should not be in the denominator.

8. POLYNOMIAL functions expressions

9. Degree of a polynomial function f(x) = -7x 3 + 6x 2 + 11x - 19 The highest exponent is 3 from -7x 3 The degree of this particular polynomial is 3

10. Types of polynomial functions 15x 2 The name of a polynomial is determined by the number of terms in it. Most common polynomials Monomials are polynomials that contain only one term . 3b 12y 4 Examples Binomials are polynomials that contain only two terms . x + y 4x + 7 9x + 2 Examples Trinomials are polynomials that contain only three terms . x 3 – 3 + 5x z 4 + 45 + 3z x 2 – 12x + 15 Examples

11. Most common types of polynomials Zero Polynomial Function is of the form f (x) = 0 Examples Linear Polynomial Function has a degree 1 f(x) = ax + b f(x) = x + 3 f(x) = 2x + 4 f(y) = 8y - 3 Quadratic Polynomial Function has a degree 2. It is of the form of f(x) = ax 2 + bx + c f(m) = 5m 2 – 12m + 4 f(x) = 14x 2 - 3 f(x) = x 2 + 4x Examples Cubic Polynomial Function has a degree 3. It is of the form of f(x) = ax 3 + bx 2 + cx + d f(y) = 4y 3 f(y) = 15y 3 – y 2 + 10 f(a) = 3a + a 3 Examples

12. POLYNOMIAL function graph A linear polynomial function is of the form y = ax + b and it represents a straight line . A quadratic polynomial function is of the form y = ax 2 + bx + c and it represents a parabola . A cubic polynomial functions is of the form y = ax 3 + bx 2 + cx + d.

13. POLYNOMIAL function graph

14. Graphs of polynomial function It is important to notice that the graphs of constant functions and linear functions are always straight lines.

15. Graphs of polynomial function Examples of quadratic functions: f(x) = x 2 f(x) = 2x 2 f(x) = 5x 2 You can see from the graph that, as the coefficient of x 2 increased , the graph is stretched vertically (that is, in the y direction.

16. Graphs of polynomial function Examples of quadratic functions: f(x) = -x 2 f(x) = -2x 2 f(x) = -5x 2

17. Graphs of polynomial function Examples of quadratic functions: What happens when we vary the coefficient of x , rather than the coefficient of x 2 . As you can see, increasing the positive coefficient of x in this polynomial moves the graph down and to the left.

18. Graphs of polynomial function Examples of quadratic functions: What happens if the coefficient of x is negative? As you can see, increasing the negative coefficient of x ( in absolute terms) moves the graph down and to the right.

19. Graphs of polynomial function Examples of quadratic functions: What happens when we vary the constant term at the end of our polynomial? As we can see , varying the constant term translates the x 2 +x curve vertically. Furthermore, the value of the constant is the point at which the graph crosses the f(x) axis.

20. Graphs of polynomial function Example of Cubic function: Draw the curve of the equation for -2 ≤ x ≤ 2 y = x 3 – 3x - 1 The y-intercept of the curve -1 and the equation ends with -1 for when x = 0.

21. Graphs of polynomial function Example of Cubic function: Draw the curve of the equation for -3 ≤ x ≤ 3 y = x 3 – 6x + 1 The y-intercept of the curve +1 and the equation ends with +1 for when x = 0.

22. Graphs of polynomial function Example of Quartic function: y = f(x) = x 4 – 5x 2 + 4 Step 1: By putting x = 0, we find the y-intercept of the function and then solve for y = f(x), we get: f(0) = (0) 4 – 5(0) 2 + 4 The y-intercept is (0,4) Step 2: Now, we find the x-intercept of the function by factoring. Considering p = x 2 , we get f(p) = p 2 – 5p + 4 = p(p-4) + 1(p-4) =(p-4)(p-1) Now, setting f(p) = 0, we get (p-4) = 0 and (p-1) = 0 p = 0 and 1

23. Graphs of polynomial function Example of Quartic function: y = f(x) = x 4 – 5x 2 + 4 Now, setting f(p) = 0, we get (p-4) = 0 and (p-1) = 0 p = 0 and 1 Placing the values of p x 2 = 4 and x 2 = 1 x = ±2 and x = ±1 The x-intercepts of the quartic parent function are (2,0), (-2,0), (1,0), and (-1,0)

24. Graphs of polynomial function Step 3: Now, selecting some points between each of the x-intercepts and evaluating the values of the function, we get

25. Graphs of polynomial function Step 4: Plotting some points between each of the x and y intercepts from step 3 and connecting the points with a curve, we get

26. Turning points of polynomial function A turning point of a function is a point where the graph of the function changes from sloping downwards to sloping upwards, or vice versa. Generally speaking, curves of degree n can have up to (n-1) turning points. The gradient changes from negative to positive, or from positive to negative.

27. Turning points of polynomial function A quartic could have up to three turning points, and so would look something like this.

28. Turning points of polynomial function Quadratic has only one turning point Cubic could have up to two turning points Some cubics have fewer turning points, for example f(x) = x 2 but no cubic has more than two turning points. A quartic could have up to three turning points, and so would look something like this.

29. Slope and y-intercept form A straight line having slope m = tan θ where θ is the angle formed by the line with the positive x-axis, and y-intercept as b is given by: y = mx + b , where m is the slope. Straight Line Passing Through a Point Equation of Straight Line

30. Linear equation graph The graph of a linear equation in one variable x forms a vertical line that is parallel to the y-axis and vice-versa, whereas, the graph of a linear equation in two variables in x and y forms a straight line . Liner Equation in two variables Example: Plot a graph for a linear equation in two variables x – 2y = 2. Step 1: Convert x – 2y = 2 in the form of y = mx + b y = x/2 - 1 Step 2: When we put x = 0 , we get y = 0/2 – 1 then y = -1. If x = 2 then y = 0. Step 3: When we x = 4 , we get y = 1. When the value of x = -2 then y = -2.

31. Linear equation graph y = f(x) Y Intercept formula Step 1 : we just substitute x = 0 Step 2 : solve for y Step 3 : represent the y-intercept as the point (0, y)

32. Linear equation graph

33. Linear equation graph Find the y-intercept of the line 3x + (-2y) = 12 First, rewrite the equation by substituting 0 for x 3(0) + (-2y) = 12 0 – 2y = 12 Because y = - 6, the y-intercept is -6.

34. Linear equation graph Graph the equation x + 2y = 7 We can find two solutions, corresponding to the x-intercepts and y-intercepts of the graph, by setting first x = 0 and then y = 0. When x = 0, we get: 0 + 2y = 7 y = 3.5 So the points are (0,3.5) and (7,0)

35. Linear equation graph y = 3x + 1 Graph the line From the equation, y-intercept is 1 Point (0,1) Slope is 3 Graph the point (0,1) and from there go up 3 units and to the right unit and graph a second point. Draw the line that contains both points.

36. Linear equation graph y = 3 x = -2 Horizontal Line Vertical Line

37. Graphing polynomial functions Let us draw the graph for the quadratic polynomial function f(x) = x 2 .

38. Graphing polynomial functions Example: x 2 – 3x – 4 = 0 (x + 1)(x - 4) x = -1 X = 4 Roots f(x) = 0 y = (0) 2 – 3(0) - 4 y = - 4

39. Dividing polynomials Is an algorithm so solve rational number that represents a polynomial divided by a monomial or another polynomial. The divisor and the dividend are placed exactly the same way as we do for regular division.

40. Dividing polynomials by monomials W hile dividing polynomials by monomials, the division can be done in two ways. Splitting the terms method . By simply separating the ‘+’ and ‘-’ operator signs and solve each part separately. Factorization Method . By factorization and further simplifying. (4x 2 – 6x) ÷ (2X) 4x 2 2x 6x 2x - 2x - 3 = (2x 2 + 4x) ÷ (2X) 2x(x + 2) 2x x + 2 =

41. Dividing polynomials by binomials For dividing polynomials by binomials or any other type of polynomials, the most common and general method is the long division method . When there are no common factors between numerator and the denominator, or if you can’t find the factors, you can use the long division process to simplify the expression.

42. Dividing polynomials by binomials Example: (4x 2 – 5x – 21) ÷ (x – 3) Dividend Divisor So, when we are dividing a polynomial (4x 2 – 5x – 21) with a binomial (x – 3), the quotient is 4x + 7 and the remainder is 0.

43. Dividing polynomials by binomials

44. Dividing polynomials using synthetic division Is a technique to divide a polynomial with a linear binomial by only considering the values of the coefficients. Step 1: Write the polynomials in the standard form from the highest degree term to the lowest degree terms. Step 2: While writing the descending powers, use 0’s as the coefficients of the missing terms. Example: (x 3 + 3) has to be written as x 3 + 0x 2 + 0x + 3

45. Dividing polynomials using synthetic division Let us divide (x 2 + 3) by x - 4 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, so the value of k is 4 .

46. Dividing polynomials using long division Example: If we were to divide 2x 3 – 3x 2 + 4x + 5 by x + 2

47. Dividing polynomials using long division Example: If we were to divide 2x 3 – 3x 2 + 4x + 5 by x + 2

48. Dividing polynomials using long division Example: If we were to divide 2x 3 – 3x 2 + 4x + 5 by x + 2

49. Dividing polynomials using long division Example: Divide 5x 2 + 3x + 2 by x + 1 Quotient 5x - 2 Remainder is

50. Dividing polynomials using long division Example: Divide 6x 3 + 11x 2 – 31x + 15 by 3x - 2

51. Dividing polynomials using long division long division simpler version

52. Dividing polynomials long division Synthetic division 2x 2 – 7x + 18 and the remainder is -31

53. Dividing polynomials using synthetic division Divide 5x 2 – 31x - 36 by x - 3 Bring down the leading coefficient. Multiply the leading coefficient by k 5x + 12 and the remainder is 0. (x - 3)(5x + 12) + 0 = 5x 2 – 3x - 36

54. Real zeros of polynomial Zeros of Polynomial are the points where the polynomial equal to zero on the whole. Are also referred to as the roots of the equation and are often designated as  , β , γ respectively. Some Methods to Find Zeros of Polynomials Grouping Factorization Algebraic expressions

55. Real zeros of polynomial The zeros of Polynomial f(x) are the values of x which satisfy the equation f(x) = 0 What are Zeros of Polynomials? The number of zeros of a polynomial depends on the degree of the equation f(x) = 0. Graphically the zeros of the polynomial are the points where the graph of y = f(x) cuts the x-axis..

56. Real zeros of polynomial (1) Linear Equation . A linear equation is of the form of y = ax + b. The zero of this equation can be calculated by substituting y = 0, and on the simplification we have ax + b = 0, or x = -b/a How to Find Zero of a Polynomial? (2) Quadratic Equation . There are two methods to factorize a quadratic equation. x 2 + x( a+b ) + ab = 0 can be factorized as ( x+a )( x+b ) = 0 x = - a, and x = - b Zeros of polynomial Quadratic formula: ax 2 + bx + c = 0, which cannot be factorized, the zeros can be calculated using quadratic formula.

57. Real zeros of polynomial (2) Quadratic Equation . The zeros of the quadratic polynomial are -3 and 5. Let  = -3 and β = 5 Example: Sam knows that the zeros of a quadratic polynomial are -3 and 5. How can we help to find the equation of the polynomial? Solution: Then we have the sum of the roots = + β = 2 product of the roots =  β = -15 The required quadratic equation is x 2 – (+ β )x +  β = 0 x 2 – 2x + (-15) = 0 x 2 – 2x – 15 = 0

58. Real zeros of polynomial (3) Cubic Equation . The cubic equation of the form y = ax 3 + bx 2 + cx + d, can be factorized by applying the remainder theorem. How to Find Zero of a Polynomial? As per the remainder theorem, we can substitute any smaller values for the variable x = , and if the value of y results to zero, y = 0, then (x-) is one root of the equation. Further, we can divide the cubic equation with (x-) using long division or synthetic division to obtain a quadratic equation. Finally, the quadratic equation can be solved either through factorization or by formula method to obtain the required roots of the equation.

59. Real zeros of polynomial (3) Cubic Equation . Example1: f(x) = x 3 + 8x 2 + 11x - 20 p ±1 ±2 ±4 ±5 ±10 ±20 q ±1 = f(1) = (1) 3 + 8(1) 2 + 11(1) - 20 f(1) = 1 + 8 + 11 - 20 f(1) = 0 x = 1 Using synthetic division 1 8 11 - 20 1 1 9 20 1 9 20 x 2 + 9x + 20 = 0 (x + 5)(x + 4) = 0 x = -5 x = -4

60. Real zeros of polynomial (3) Cubic Equation . Example 2: f(x) = x 3 - 11x + 6 p ±1 ±2 ±3 ±6 q ±1 = f(1) = (1) 3 - 11(1) + 6 = -4 f(2) = (2) 3 -11(2) + 6 = -8 f(3) = (3) 3 -11(3) + 6 = 0 x = 3 Using synthetic division 1 0 -11 6 1 3 9 -6 1 3 -2 0 x 2 + 3x - 2 = 0 Next, use quadratic formula

61. Real zeros of polynomial (3) Cubic Equation . Example 2: f(x) = x 3 - 11x + 6 p ±1 ±2 ±3 ±6 q ±1 = f(1) = (1) 3 - 11(1) + 6 = -4 f(2) = (2) 3 -11(2) + 6 = -8 f(3) = (3) 3 -11(3) + 6 = 0 x = 3 Using quadratic formula x 2 + 3x - 2 = 0 a = 1 b = 3 c = -2 x = -b ±√b 2 – 4ac 2a -3 ±√3 2 – 4(1)(-2) 2(1) -3 +√17 2 x = -3 -√17 2 x =

62. Real zeros of polynomial (3) Cubic Equation . Example 2: f(x) = x 3 + 2x 2 – 5x - 6 p ±1 ±2 ±3 ±6 q ±1 = f(1) = (1) 3 + 2(1) 2 – 5(1) – 6 = -8 f(2) = (2) 3 + 2(2) 2 – 5(2) – 6 = 0 x = 2 Using synthetic division 1 2 -5 -6 2 2 8 6 1 4 3 0 x 2 + 4x + 3 = 0 (x + 3)(x + 1) = 0 x = -3 & x = -1

63. Real zeros of polynomial (3) Cubic Equation . Example: f(x) = x 3 - 12x 2 + 20x Solution: Let us take out x as common f(x) = x(x 2 - 12x + 20) Now by splitting the middle term f(x) = x(x 2 – 2x - 10x + 20) So, we get f(x) = x[x(x – 2) – 10(x - 2)] f(x) = x(x – 2)(x - 10)] x = 0 x = 2 x = 10 Zeros of polynomial function

64. representing zeros of polynomial on graph

65. Important notes on zeros of polynomial The zeros of polynomial are the values of the variable for which the polynomial is equal to 0. We can find the zeros of polynomial by determining the x-intercepts. To find zeros of a quadratic polynomial, we use the quadratic formula.

66. Worksheet X + 2y = 7 X + 2y = 7