POLYNOMIALS by: Ms. P

Today’s Objectives:  Review Classify a polynomial by it’s degree.  Review complete a square for a quadratic equation and solve by completing the square

Degree of a Polynomial The degree of a polynomial is calculated by finding the largest exponent in the polynomial. (Learned in previous lesson on Writing Polynomials in Standard Form)

Degree of a Polynomial (Each degree has a special “name”) 9

9 No variable

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degree

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degree

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degree

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic 3x 4 + 5x – 1

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic 3x 4 + 5x – 1 4 th degree

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic 3x 4 + 5x – 1 4 th degreeQuartic

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic 3x 4 + 5x – 1 4 th degreeQuartic 2x 5 + 7x 3

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic 3x 4 + 5x – 1 4 th degreeQuartic 2x 5 + 7x 3 5 th degree

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic 3x 4 + 5x – 1 4 th degreeQuartic 2x 5 + 7x 3 5 th degreeQuintic

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic 3x 4 + 5x – 1 4 th degreeQuartic 2x 5 + 7x 3 5 th degreeQuintic 5x n

Degree of a Polynomial (Each degree has a special “name”) 9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic 3x 4 + 5x – 1 4 th degreeQuartic 2x 5 + 7x 3 5 th degreeQuintic 5x n “nth” degree

9 No variableConstant 8x 1 st degreeLinear 7x 2 + 3x 2 nd degreeQuadratic 6x 3 – 2x 3 rd degreeCubic 3x 4 + 5x – 1 4 th degreeQuartic 2x 5 + 7x 3 5 th degreeQuintic 5x n “nth” degree Degree of a Polynomial (Each degree has a special “name”)

Let’s practice classifying polynomials by “degree”. POLYNOMIAL 1.3z 4 + 5z 3 – a c 10 – 7c 6 + 4c f 3 – 7f y 2 7.9g 4 – 3g r 5 –7r 9.16n 7 + 6n 4 – 3n 2 DEGREE NAME 1.Quartic 2.Linear 3.Constant 4.Tenth degree 5.Cubic 6.Quadratic 7.Quartic 8.Quintic 9.Seventh degree The degree name becomes the “first name” of the polynomial.

- Completing the Square Objective: To complete a square for a quadratic equation and solve by completing the square

Steps to complete the square 1.) You will get an expression that looks like this: AX²+ BX 2.) Our goal is to make a square such that we have (a + b)² = a² +2ab + b² 3.) We take ½ of the X coefficient (Divide the number in front of the X by 2) 4.) Then square that number

To Complete the Square x 2 + 6x Take half of the coefficient of ‘x’ Square it and add it 3 9 x 2 + 6x + 9 = (x + 3) 2

Complete the square, and show what the perfect square is:

To solve by completing the square If a quadratic equation does not factor we can solve it by two different methods 1.) Completing the Square (today’s lesson) 2.) Quadratic Formula (Next week’s lesson)

Steps to solve by completing the square 1.) If the quadratic does not factor, move the constant to the other side of the equation Ex: x²-4x -7 =0 x²-4x=7 2.) Work with the x²+ x side of the equation and complete the square by taking ½ of the coefficient of x and squaring Ex. x² -4x 4/2= 2²=4 3.) Add the number you got to complete the square to both sides of the equation Ex: x² -4x +4 = )Simplify your trinomial square Ex: (x-2)² =11 5.)Take the square root of both sides of the equation Ex: x-2 =±√11 6.) Solve for x Ex: x=2±√11

Solve by Completing the Square +9

Solve by Completing the Square +121

Solve by Completing the Square +1

Solve by Completing the Square +25

Solve by Completing the Square +16

Solve by Completing the Square +9

The coefficient of x 2 must be “1”

Do you remember : How is the highest (maximum) or lowest point (minimum) of a quadratic function found?

Using the completing the square form for quadratics: