review of long division 6.4 – dividing polynomials long division

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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.