Math 2B/3A

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QuadraticsQuiz1Review–FactoringandSolvingQuadraticEquationsStudentsWillBeAbleto:

• Usevocabularytodescribepolynomialexpressions• Factorpolynomialsthathaveacommonfactor,includingabinomialcommonfactor.• FactorpolynomialsusingDifferenceofSquaresfactoring• FactorquadraticsusingSplittingtheMiddleTerm• Factorpolynomialsthatrequireusingmorethanonesteporstrategy• Factorpolynomialsbycompletingthesquare• SolvequadraticequationsusingtheZeroProductProperty(ZPP)• SolvequadraticequationsusingtheQuadraticFormula• ReviewTopic:Findcompositionsoffunctions

VocabularyfromthisUnit:

binomial discriminant monomial solvecoefficient factor nthdegreepolynomial standardformconstantterm greatestcommonfactor nthdegreeterm termcubicpolynomial leadingterm polynomial trinomialcubicterm linearpolynomial quadraticpolynomial degree linearterm quadraticterm

PracticeProblems:Allworkmustbeshowntoearnanycredit!

1. Usethefollowingfunctionstofindthegivencompositions.Simplifyfully.

𝑔 𝑥 = 3𝑥 + 5 𝑓 𝑥 = 𝑥! − 6

a. 𝑓(𝑔 𝑥 ) b. 𝑔(𝑓 𝑥 )

Math 2B/3A

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2. Usethepolynomial𝑝 𝑥 = −6𝑥! + 5𝑥 − 2𝑥! + 4− 3𝑥

a. Write𝑝(𝑥)instandardform.

b. Whatistheleadingcoefficient?

c. Whatisthedegreeof𝑝(𝑥)?

d. Find𝑟(𝑥)suchthat𝑝 𝑥 − 𝑟 𝑥 = −3𝑥! −5𝑥! + 2𝑥 + 5

e. Find𝑞 𝑥 suchthat𝑝 𝑥 + 𝑞 𝑥 = 2𝑥 + 4

f. Whatisthequadraticcoefficientof𝑝(𝑥)instandardform?

g. Name𝑞(𝑥)and𝑟(𝑥)basedontheirnumberofterms.q(x):r(x):

3. Giventhepolynomial𝑓 𝑥 = 3𝑥! − 4𝑥,findapolynomialthatfitseachdescription.

a. Apolynomial𝑔(𝑥)suchthat𝑓 𝑥 ∗ 𝑔(𝑥)

hasdegree5.

b. Apolynomial𝑘(𝑥)suchthat𝑓 𝑥 + 𝑘(𝑥)hasdegree3.

Math 2B/3A

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𝑓 𝑥 = 3𝑥! − 4𝑥c. Apolynomial𝑚(𝑥)suchthat𝑓 𝑥 −𝑚 𝑥 = 6

d. Whatisthelineartermof𝑓(𝑥)?

e. Whatistheconstanttermof𝑚(𝑥)?

4. Writeapolynomialinonevariablethatfitseachdescription.

a. Alinearbinomialwithaconstantof−4

instandardform.

b. Aquadratictrinomialwithalinearcoefficientof3.

c. A6thdegreemonomial.

d. A4thdegreebinomialwithnocubicterminstandardform.

5. FactoreachofthefollowingusingGCFfactoring.

a. 2𝑥 3𝑥 − 4 − 7 3𝑥 − 4

b. 𝑥 2𝑥 + 1 − (2𝑥 + 1)

c. 12𝑥! − 4𝑥!

d. 5𝑥!𝑦 + 15𝑥𝑦! − 20𝑥𝑦

Math 2B/3A

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6. FactoreachofthefollowingusingSplittingtheMiddleTermFactoringa. 𝑥! − 6𝑥 − 7

b. 𝑥! + 2𝑥 − 35

7. FactoreachofthefollowingusingDifferenceofSquares 𝑎 + 𝑏 𝑎 − 𝑏 = 𝑎! − 𝑏!

a. 49𝑦! − 64

b. (𝑚 + 4)! − 25

c. 121− 16𝑝!

d. 81𝑘! − 100𝑚!

e. Whatcharacteristicsdoalloftheseexpressionshaveincommon?

8. FactoreachofthefollowingusingSplittingtheMiddleTerm

a. 6𝑥! − 7𝑥 − 5

b. 5𝑥! − 33𝑥 + 18

Math 2B/3A

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9. FactoreachofthefollowingbyCompletingtheSquare.a. 𝑥! − 6𝑥 − 112

b. 𝑥! + 12𝑥 − 108

10. Fullyfactoreachofthefollowingusinganefficientmethod.Namethemethodused.

a. 𝑥! + 12𝑥 + 27 b. 𝑥! − 8𝑥 − 105

c. 2𝑥! − 5𝑥! d. 5𝑥! + 14𝑥 − 3

e. 25𝑥! − 144

f. 𝑥! − 28𝑥 + 192

Math 2B/3A

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11. Factoreachofthefollowingcompletely.Youwillneedtousemorethanonetechnique.

a. 5𝑥! + 15𝑥! − 20𝑥

b. 12𝑥! − 7𝑥! − 10𝑥

c. 16𝑥! − 1

d. 2𝑥! − 28𝑥 + 98

e. 75𝑥! − 27𝑥

f. 𝑥! + 10𝑥! − 24𝑥

12. UsethediscriminanttodeterminehowmanyRealNumbersolutionseachofthe

equationshas.Youdonotneedtosolve.a. 9𝑥! + 12𝑥 + 4 = 0

b. 3𝑥! + 4𝑥 + 12 = 0

Math 2B/3A

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13. SolveeachofthefollowingequationsusingtheQuadraticFormula.

a. 6𝑥! + 𝑥 − 35 = 0 b. 9𝑥! − 9𝑥 = −5

c. 𝑥! − 8 = −6𝑥 d. 0 = 𝑥! + 4𝑥 + 1

Math 2B/3A

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Solveeachofthefollowingequations.Useanefficientmethod.a. 𝑥! − 5𝑥 = 14

b. 𝑥! − 40 = −3𝑥

c. 𝑥! + 4𝑥 = 165

d. 36𝑥! = 25

Math 2B/3A

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Chapter3Quiz1Review–AnswerstoPracticeProblems1.

a. 9𝑥! + 30𝑥 + 19b. 3𝑥! − 13

2. a. 𝑝 𝑥 = −6𝑥! − 2𝑥! + 2𝑥 + 4b. −6c. 3d. 𝑟 𝑥 = −3𝑥! + 3𝑥! − 1e. 𝑞 𝑥 = 6𝑥! + 2𝑥!f. −2g. 𝑞(𝑥)isabinomialand𝑟(𝑥)isatrinomial

3. a. Answersvary,𝑔(𝑥)musthavedegree3.b. Answersvary,𝑘(𝑥)musthavedegree3.c. 𝑚 𝑥 = 3𝑥! − 4𝑥 − 6d. −4𝑥e. −6

4. Answersvary,possibleanswersaregiven

a. 𝑥 − 4b. 𝑥! + 3𝑥 + 1c. 4𝑥!d. 𝑥! + 2𝑥!

5. a. 3𝑥 − 4 2𝑥 − 7 b. 2𝑥 + 1 𝑥 − 1 c. 4𝑥!(3𝑥 − 1)d. 5𝑥𝑦 𝑥 + 3𝑦 − 4

6. a. 𝑥 − 7 𝑥 + 1 b. 𝑥 + 7 𝑥 − 5

7. a. 7𝑦 + 8 7𝑦 − 8 b. 𝑚 + 4 + 5 𝑚 + 4 − 5 =

𝑚 + 9 𝑚 − 1 c. (11 + 4𝑝)(11 − 4𝑝)d. (9𝑘! + 10𝑚)(9𝑘! − 10𝑚)

e. Theyarebinomialswherebothtermsareprefectsquaresandthesecondtermisbeingsubtracted.

8. a. 2𝑥 + 1 3𝑥 − 5 b. (𝑥 − 6)(5𝑥 − 3)

9. a. (𝑥 − 14)(𝑥 + 8)b. 𝑥 + 10 𝑥 + 11

10. a. SplittingtheMiddleTerm 𝑥 + 3 𝑥 + 9 b. CompletingtheSquare 𝑥 − 15 𝑥 + 7 c. GCFFactoring𝑥!(2𝑥 − 5)d. SplittingtheMiddleTerm(5𝑥 − 1)(𝑥 + 3)e. Diff.ofSquares(5𝑥! + 12)(5𝑥! − 12)f. CompletingtheSquare(𝑥 − 16)(𝑥 − 12)

11. a. 5𝑥(𝑥 + 4)(𝑥 − 1)b. 𝑥(3𝑥 + 2)(4𝑥 − 5)c. 4𝑥! + 1 2𝑥 + 1 2𝑥 − 1 d. 2 𝑥 − 7 𝑥 − 7 or2(𝑥 − 7)!e. 3𝑥 5𝑥 + 3 5𝑥 − 3 f. 𝑥 𝑥 + 12 𝑥 − 2

12. a. oneRealsolutionb. noRealsolutions

13. a. − !

!, !!

b. Norealsolutionsc. 𝑥 = −3 ± 17d. 𝑥 = −2 ± 3

14. a. 𝑥 = −2,−7b. 𝑥 = −8,−5c. 𝑥 = −15, 11d. 𝑥 = ± !

!