6 .4 – dividing polynomials
DESCRIPTION
6 .4 – Dividing Polynomials. Long Division. Review of Long Division. 6.4 – Dividing Polynomials. Long Division. 6.4 – Dividing Polynomials. Long Division. 6.4 – Dividing Polynomials. Long Division. 5.7 - Factoring Polynomials. 20 is the result of the product of 4 and 5. - PowerPoint PPT PresentationTRANSCRIPT
285
5
Review of Long Division
783 4
4 7835 2855
2535
7
350
438362
19
3
5
203
6.4 – Dividing PolynomialsLong Division
4
3195
4
3195
72812 2 xxx
x4
xx 48 2 x6 7
3
36 x
4
12 x
12
434
xx 728 2 xx
12
4
x
6.4 – Dividing PolynomialsLong Division
12
728 2
x
xx
35125 2 xxx
x
xx 52 x7 35
7
357 x
0
5x 7x 35122 xx
6.4 – Dividing PolynomialsLong Division
5
35122
x
xx
15023 2 xxx
x2
xx 62 2 x6 15
6
186 x
3
3x
3
362x
x 152 2 x
3
3
x
6.4 – Dividing PolynomialsLong Division
3
152 2
x
x
5.7 - Factoring Polynomials
Factoring Trinomials
4 5x x
2x bx c 2 5 4 20x x x 2 9 20x x
4 5x x 2 209x x
20 is the result of the product of 4 and 5.
9 is the result of the sum of 4 and 5.
Factors of 20 are: 1, 20 2,10 4, 5
Factoring Trinomials 2x bx c 2 10 24x x
Factors of 24 are: 1, 24 2, 12 3, 8 4, 6
x x
4 6x x
5.7 - Factoring Polynomials
Factoring Trinomials 2x bx c 2 27 50x x
Factors of 50 are: 1, 50 2, 25 5,10
x x
2 25x x
5.7 - Factoring Polynomials
Factoring Trinomials 2x bx c
Factors of 9 are: 1, 9 3, 3
4 x x
4 3 3x x
24 24 36x x
24 6 9x x
5.7 - Factoring Polynomials
Factoring Trinomials 2ax bx c
Factors of 12 are: 1, 12 2,6 3, 4
x x
4 2 3x x
22 11 12x x
Factors of 2 are: 1, 2
5.7 - Factoring Polynomials
Factoring Perfect Square Trinomials
6 6x x
2 12 36x x
Not a perfect square
36132 xx
94 xx
26x
x x x x
5.7 - Factoring Polynomials
Factoring Perfect Square Trinomials 2 18 81x x
9 9x x
29 42 49x x
3 7 3 7x x
29x 273 x
x x x x
5.7 - Factoring Polynomials
Factoring the Difference of Two Squares
3 1 3 1s s 29 1s
2 81p
24 100x
9 9p p
Not the difference
5.7 - Factoring Polynomials
Factoring the Difference of Two Squares
3 38 8
5 5c c
2 9
6425
c
2 2121 49x y 11 7 11 7x y x y
5.7 - Factoring Polynomials
Factoring the Sum or Difference of Two Cubes5.7 - Factoring Polynomials
2233
2233
babababa
babababa
273 xx 3 2x x3 9
Factoring the Sum or Difference of Two Cubes5.7 - Factoring Polynomials
2233
2233
babababa
babababa
83 yy 2 2y y2 4
Factoring the Sum or Difference of Two Cubes5.7 - Factoring Polynomials
2233
2233
babababa
babababa
33 64yx
x 4 2x xy4 16 y 2y
Factoring the Sum or Difference of Two Cubes5.7 - Factoring Polynomials
2233
2233
babababa
babababa
23227 aba
3 9 b3 b2b
2a 327 b
2a
A quadratic equation is written in the Standard Form, 2 0ax bx c where a, b, and c are real numbers and .0a
5.8 – Solving Equations by Factoring
Zero Factor Property: 0ab
then or . 0a 0b If a and b are real numbers and if ,
Zero Factor Property: If a and b are real numbers and if , 0ab
Examples: 10 3 6 0x x
then or . 0a 0b
10 0x 3 6 0x
10x 3 6x 2x
10 10 01 0x 63 66 0x 3 6
3 3
x
5.8 – Solving Equations by Factoring
Solving Equations by Factoring: 1) Write the equation to equal zero.
4) Solve each equation.
2) Factor the equation completely.
3) Set each factor equal to 0.
5) Check the solutions (in original equation).
5.8 – Solving Equations by Factoring
2 3 18 0x x
6 0x 3 0x 3x
6x 3x
2 3 18x x
18 :Factors of1,18 2, 9 3, 6
26 3 16 8
36 18 18
18 18
213 3 83
9 9 18 18 18
6x 0
5.8 – Solving Equations by Factoring
3 18x x 18x
2 3 18x x
218 13 18 8
324 54 18
270 18
221 23 11 8
441 63 18 378 18
3 18x
3 183 3x 21x
If the Zero Factor Property is not used, then the solutions will be incorrect
5.8 – Solving Equations by Factoring
2 4 5x x
1 0x 5 0x
1 5 0x x
1x 5x
4 5x x
2 4 5 0x x
5.8 – Solving Equations by Factoring
23 7 6x x 3 0x 3 2 0x
3 3 2 0x x
3x 2
3x
3 7 6x x
23 7 6 0x x 3 2x
6 :Factors of2, 31, 6
3:Factors of1, 3
5.8 – Solving Equations by Factoring
29 24 16x x 29 24 16 0x x
3 4 0x 3 4 3 4 0x x
4
3x
3 4x
9 16and are perfect squares
5.8 – Solving Equations by Factoring
32 18 0x x 2x
2 0x
2x
3x 3 0x 3 0x
3x 0x
2 9x 0
3x 3x 0
5.8 – Solving Equations by Factoring
23 3 20 7 0x x x
3x
3 0x
7x 7 0x 3 1 0x
1
3x
3x 3 1x
3:Factors of 1, 3 7 :Factors of 1, 7
7x 0 3 1x
5.8 – Solving Equations by Factoring
0
A cliff diver is 64 feet above the surface of the water. The formula for calculating the height (h) of the diver after t seconds is: 216 64.h t How long does it take for the diver to hit the surface of the water?
0 0
2 0t 2 0t 2t 2t seconds
216 64t 16 2 4t
16 2t 2t
5.8 – Solving Equations by Factoring
5.8 – Solving Equations by Factoring
2x
The square of a number minus twice the number is 63. Find the number.
7x
7x
x is the number.
2 2 63 0x x
7 0x 9 0x
9x
2x 63
63:Factors of 1, 63 3, 21 7, 9
9x 0
5 176w w
The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. What are the length and the width of the garden?
11w The width is w.
11 0w 11w
The length is w+5.l w A
2 5 176w w 2 5 176 0w w
16 0w 16w
11w 11 5l 16l
feet
feet
176 :Factors of1,176 2, 88 4, 44
8, 22 11,16
16w 0
5.8 – Solving Equations by Factoring
x
Find two consecutive odd numbers whose product is 23 more than their sum?
Consecutive odd numbers: x
5x 5x 2 2 2 25x x x
2 25 0x 5x
5 0x 5 0x
5, 3 5, 7
5 2 3 5 2 7
2.x 2x 2x x 23
2 22 2 2 25xx x x x
2 25 2525x
2 25x
5x 0
5.8 – Solving Equations by Factoring
a x
The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 meters. What are the lengths of the legs?
12a
.Pythagorean Th
22 27 13x x
5x
5
meters
7b x 13c
2 2 14 49 169x x x 22 14 120 0x x
22 7 60 0x x
2
5 0x 12 0x
12x
12 7b meters
2 2 2a b c
60 :Factors of 1, 60 2, 303, 20 4,15 5,12
5x 12x 0
6,10
5.8 – Solving Equations by Factoring
6.1 – Rational Expressions -
A rational expression is a quotient of polynomials.
For any value or values of the variable that make the denominator zero, the rational expression is considered to be undefined at those value(s).
3
5 1
x
x
23 6 7
2 6
x x
x
4
3
4 4 12
q
q q The denominator can not equal zero.
Multiplying and Dividing
What are the values of the variable that make the denominator zero and the expression undefined?
3
5 1
x
x
23 6 7
2 6
x x
x
2
3 2
12
x
x x
5 1 0x
5 1x 1
5x
2 6 0x
2 6x 3x
3 4 0x x
2 12 0x x
3 0x 4 0x 3x 4x
6.1 – Rational Expressions – Mult. And Div.
Simplifying4 3
5 5
x x
x
2
5
25
q
q
3x
3
5
x
5q
1
5q
1x
5 1x 5q 5q
6.1 – Rational Expressions – Mult. And Div.
Simplifying2
2
11 18
2
x x
x x
2x
9
1
x
x
9x
2x 1x
6.1 – Rational Expressions – Mult. And Div.
Simplifying4
4
x
x
4x
4x
4x
1 4x 1
6.1 – Rational Expressions – Mult. And Div.
6.1 - Rational Expressions – Mult. And Div.
3 2
3
5 2
3 15
a b
b a
10
2
3a 2b45 a 3b
2a9 b
Multiplication:
6.1 - Rational Expressions – Mult. And Div.
3 2
3
5 2
3 15
a b
b a
2 2a9 b
Multiplication: 1
3
2
1
Multiplication: 2
3 2
3 6 7
14 2
x x
x x
3
21
2x 2x
27x14 2x
1432
6.1 - Rational Expressions – Mult. And Div.
19
253
147
842
2
2
x
xx
xx
x
4 2x
7x 2x 3 1x
3 1x 3 1x 2x
4 2x
7x 3 1x
Multiplication:
6.1 - Rational Expressions – Mult. And Div.
y
xx
26
7 2
27
6
x 1
3
7xy
2y
x
1
3
Division:
6.1 - Rational Expressions – Mult. And Div.
2
123
6
4 2
xx
24
6
x
24
6
x
4
3
x 9
4x
2
3 12x
23 4x
1
1
3
1
3
Division:
6.1 - Rational Expressions – Mult. And Div.
2
25
4
410 23
2
x
xx
x
x
2
2
10 4
4
x
x
2
2x 2
22 xx
3 2
2
5 2
x
x x
5 2x
2x 2x 2x
2x 5 2x
2
1
x
Division:
6.1 - Rational Expressions – Mult. And Div.
21
129
147
8103 2
x
x
xx
21
21
23 10 8
7 14
x x
x
2x
1
7
21
9 12x
3 4x
7 2x 21
3 3 4x
21
31
Division:
6.1 - Rational Expressions – Mult. And Div.