POLYNOMIALSPOLYNOMIALS

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

9

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

9 No variable

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

9 No variable Constant

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

9 No variable Constant

8x

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

9 No variable Constant

8x 1st degree

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

9 No variable Constant

8x 1st degree Linear

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

3x4 + 5x – 1

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

3x4 + 5x – 1 4th degree

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

3x4 + 5x – 1 4th degree Quartic

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

3x4 + 5x – 1 4th degree Quartic

2x5 + 7x3

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

3x4 + 5x – 1 4th degree Quartic

2x5 + 7x3 5th degree

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

3x4 + 5x – 1 4th degree Quartic

2x5 + 7x3 5th degree Quintic

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

3x4 + 5x – 1 4th degree Quartic

2x5 + 7x3 5th degree Quintic

5xn

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

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

3x4 + 5x – 1 4th degree Quartic

2x5 + 7x3 5th degree Quintic

5xn “nth” degree

9 No variable Constant

8x 1st degree Linear

7x2 + 3x 2nd degree Quadratic

6x3 – 2x 3rd degree Cubic

3x4 + 5x – 1 4th degree Quartic

2x5 + 7x3 5th degree Quintic

5xn “nth” degree “nth” degree

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

Let’s practice classifying polynomials by “degree”.

POLYNOMIAL1. 3z4 + 5z3 – 72. 15a + 253. 1854. 2c10 – 7c6 + 4c3 - 95. 2f3 – 7f2 + 16. 15y2

7. 9g4 – 3g + 58. 10r5 –7r9. 16n7 + 6n4 – 3n2

DEGREE NAME1. Quartic2. Linear3. Constant4. Tenth degree5. Cubic6. Quadratic7. Quartic8. Quintic9. 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 Squarex2 + 6x

•Take half of the coefficient of ‘x’ •Square it and add it3

9

x2 + 6x + 9

= (x + 3)2

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

xx 122 36122 xx 26x

yy 142 49142 yy 27y

yy 102 25102 yy 25y

xx 52 4

2552 xx

2

2

5

x

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 theconstant 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 coefficientof 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 equationEx: x² -4x +4 = 7 +4

4.)Simplify your trinomial square Ex: (x-2)² =11

5.)Take the square root of both sides of the equationEx: x-2 =±√11

6.) Solve for xEx: x=2±√11

Solve by Completing the Square2 6 16 0x x

2 6 16x x +9 +9

2 6 9 25x x 2

3 25x 3 5x

3 5x 8x 2x

Solve by Completing the Square

2 22 21 0x x 2 22 21x x

+121 +1212 22 121 100x x

211 100x

11 10x 11 10x 21x 1x

Solve by Completing the Square

2 2 5 0x x 2 2 5x x

+1 +12 2 1 6x x

21 6x

1 6x 1 6x

Solve by Completing the Square

2 10 4 0x x 2 10 4x x

+25

+252 10 25 29x x

25 29x 5 29x

5 29x

Solve by Completing the Square

01182 xx1182 xx

+16

+16 51682 xx

54 2 x 54 x

54 x

Solve by Completing the Square

0462 xx462 xx

+9 +9

5962 xx 53 2 x 53 x

53x

The coefficient of x2 must be “1”

0332 2 xx

02

3

2

32 xx16

33

4

32

x

2

3

2

32 xx4

32

2

3

2 9 9

16 16

3 3

2 2x x

16

33

4

3x

16

33

4

3x

2 2 2 2

33

44

333x

The coefficient of x2 must be “1”23 12 1 0x x

2 14

3x x

2 4 41

43

x x

2 112

3x

112

3x

112

3x 3

3

332

3x

332

3x

6

3

6 33

3x

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: