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Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

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Section 5-4 Factor and solve Polynomial Equations

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Page 1: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Quiz 5-31. Simplify:1. Simplify:

3.3. Simplify (special pattern):Simplify (special pattern):

2. Simplify:2. Simplify:

)52(442 535 xxxxx

?)4)(4( 35 xxxx

?)42)(42( xx

Page 2: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Section 5-4Factor and solve Polynomial

Equations

Page 3: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

VocabularyPrime PolynomialPrime Polynomial: A polynomial where: A polynomial where the the coefficientcoefficient of each term has no common of each term has no common factor with any other coefficients factor with any other coefficients andand……

32 23 xxx

(There is no number (other than +1 or -1)(There is no number (other than +1 or -1) that is common to that is common to allall of these terms.) of these terms.)

This is important because if you can factor out the common This is important because if you can factor out the common term, it will end up being easier to factor the polynomial.term, it will end up being easier to factor the polynomial.

it cannot be factored into polynomials of it cannot be factored into polynomials of lesser degree.lesser degree.

Page 4: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

NOT a prime polynomial

31263 23 xxx

33 is a common factor of each term is a common factor of each term

We can “factor out” the 3We can “factor out” the 3

)142(3 23 xxx

Page 5: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Prime vs. Not Prime Polynomials:

ExampleExample: is : is notnot prime because it prime because it cancan be factored. be factored.

42 x

)2)(2(42 xxx

ExampleExample: is prime because it : is prime because it cannotcannot be factored. be factored.

42 x

Page 6: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Your Turn:Are these prime polynomials or not? If not,Are these prime polynomials or not? If not, write it in factored form.write it in factored form.

122 xx1. 1.

2.2. 42 2 x

Page 7: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

More Vocabulary

Factored completelyFactored completely: a polynomial is : a polynomial is factored completely if it is written as the factored completely if it is written as the product of a product of a monomialmonomial and one or more and one or more prime polynomials.prime polynomials.

ExampleExample: : ?363 2 xx

)12(3 2 xx

Can this be factored?Can this be factored?

)1)(1(3 xx

Page 8: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Your Turn:

16164 2 xx3. Factor this polynomial completely3. Factor this polynomial completely

Page 9: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Move Vocabulary

Factor by groupingFactor by grouping: some polynomials can be: some polynomials can be factored easily if they have pairs of terms thatfactored easily if they have pairs of terms that have a common monomial factor.have a common monomial factor.

1892 23 xxx

)2(2 xx

What’s the What’s the common monomialcommon monomial factor here?factor here?

What’s the What’s the common monomialcommon monomial factor here?factor here?

)2(9 x

)2)(9( 2 xx

Are both of theseAre both of these factors prime?factors prime?

)2)(3)(3( xxxDo these two expressionsDo these two expressions have a common factor?have a common factor?

Page 10: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Hints on factoring by grouping

• You need two pairs of terms in order to factor by grouping

• If they give you a 4 term polynomial, try factoring by grouping.

Page 11: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Your turn:

(problems 2 and 3 in the green book)

4. Factor 4. Factor 50252 23 xxx

Page 12: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Special Patterns (we learned these last time)

))(( 2233 xxyxyxyx Sum of 2 “cubes”Sum of 2 “cubes”

Difference of 2 “cubes”Difference of 2 “cubes”

))(( 2233 xxyxyxyx Difference of 2 “squares”Difference of 2 “squares”

)52)(52(254 2 xxx

Page 13: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Your turn:

Factor completely (special pattern)Factor completely (special pattern)

5. 5. 648 3 x

6.6.24 481 xx

(prob. #1 pg 143 green book)(prob. #1 pg 143 green book)

Page 14: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

What does solving a polynomial mean?

Where does the polynomial cross the x-axis.Where does the polynomial cross the x-axis.

35182222 234 xxxxy

Page 15: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Solve by factoring

Solve Solve rewrite the equation so that it = 0 rewrite the equation so that it = 0

Factor the polynomialFactor the polynomial

Use the Use the Zero product ruleZero product rule to find the solution. to find the solution.

24 109 xx

0910 24 xx

0)1)(9( 22 xx0)1)(1)(3)(3( xxxx

x = 3, -3, 1, -1x = 3, -3, 1, -1 How many times does this How many times does this polynomial cross the x-axis?polynomial cross the x-axis?

Page 16: Quiz 5-3 1. Simplify: 2. Simplify: Simplify (special pattern):

Your turn:

7. Solve the polynomial.7. Solve the polynomial.

35 14242 xxx

(problem #4 on green book page 143)(problem #4 on green book page 143)