5.8 1

5.8 Polynomial Equations - Solving Equations by Factoring Solutions to Equations in One Variable:

Where the curve crosses the x-Axis (if it does!) Solving Quadratic Equations

ax2 + bx + c = 0 Zero factor property

Solving Higher Degree Equations Problem Solving Not covered:

Functions/Graphs

5.8 2

Curves and x-Intercepts y=x+3 y=x2 y=x3 y=|x|

5.8 3

Definition

Zero Factor Property

5.8 4

Solving a Quadratic Equation by Factoring

5.8 5

solutionsrationalNorationalsoverfactorCannot

x

x

03

32

2

75

34 &

057&0430)57)(43(

xxxx

xx

solutionOnlyxandx

xxx

144

0)4(0168

2

2

23

43

2

&0)32)(34(

6126720968

9)34(2

yyyy

yysobacyy

yy

Examples: 0, 1, or 2 solutions

RewriteFactorSet Each = 0Solve EachCheck

Unfactorable quadratics are prime and have no rational number solutions.

5.8 6

solutionsrationalNoRationalsoverfactorableNot

x

x

07

72

2

2&0)2)(16(

02116

2)42(3

61

2

xxxx

factoringerrorandtrialxx

xxx

43&3034&03

0)34)(3(

xxxx

xx

solutionOnlyzz

zzz

11&1

0)1(012

2

2

Practice: 0, 1, or 2 solutions

RewriteFactorSet Each = 0Solve EachCheck

Unfactorable quadratics are prime and have no rational number solutions.

5.8 7

Higher Degree Polynomial Equations Write in descending powers of the variable = 0 Make leading coefficient positive (multiply by -1) Factor Solve each factor = 0 May have 0 to n (the degree) solutions Check all solutions in original equation

5.8 83,20)3)(3)(2)(2(

0)9)(4(0361301336

22

24

24

aaaaaa

aaaa

aa

1sec

0)1)(1(

012

3

xsolutionOneprimeisfactorondthe

xxx

x

Examples, 0 to n solutions

RewriteFactorSet Each = 0Solve EachCheck

Unfactorable quadratics are prime and have no rational number solutions.

3,,00)3)(25(0)6135(

6135

52

2

23

xxxxxxxxx

xxx

5.8 9

2sec

0)42)(2(

082

3

bsolutionOneprimeisfactorondthe

bbb

b

Practice, 0 to n solutions

RewriteFactorSet Each = 0Solve EachCheck

Unfactorable quadratics are prime and have no rational number solutions.

2,2,00)2)(2(2

0)4(26682

2

3

xxxxxx

xxxx

5,20)5)(5)(2)(2(

0)25)(4(010029029100

22

24

24

xxxxxx

xxxx

xx

5.8 10

Integers Consecutive Integers:

n, n+1, n+2, n+3, … Consecutive

Even or Odd Integers:n, n+2, n+4, n+6, …

Find 3 consecutive odd integers where the product of the first 2 is 68 more than the 3rd integer.

5,7,9,6363685)7(9

'8980)9)(8(

072

722

684)2(

2

2

so

oddtisnnandn

nnnn

nnn

nnn

5.8 11'5611'12618

3)(5.11)3)(232(20)69292(20

1385840

4223619860

)211)(218(60)(

2

2

2

LWborfalseb

bbbb

bb

bbb

bbwidthborderbWLA

Borders

5.8 12

Gravity

SLINGSHOT A slingshot can provide an initial velocity of 128 feet per second. At what times will a stone, shot vertically upward, be 192 feet above the ground? sandtimesAt

tttt

tt

tt

tvth

sec'620)6)(2(160)128(16

019212816

16128192

16

2

2

2

2

5.8 13

What Next? 6.1 Rational Expressions

Look for patterns …