Post on 26-Dec-2015
transcript
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 Square
2 22 21 0x x 2 22 21x x
+121 +1212 22 121 100x x
211 100x
11 10x 11 10x 21x 1x
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
Do you remember :
• How is the highest (maximum) or lowest point (minimum) of a quadratic function found?