Notes21.1a

SolvingEquationsbyFactoring𝒙𝟐 + 𝒃𝒙+ 𝒄(part1)Rememberthatthequadraticexpression;𝑥! + 𝑏𝑥 + 𝑐istheproductoftwopolynomialfactors 𝑥 + 𝑑 𝑥 + 𝑒 .

𝑥 + 𝑑 𝑥 + 𝑒 = 𝑥! + 𝑏𝑥 + 𝑐Forexample: 𝑥 + 5 𝑥 + 2 = 𝑥! + 7𝑥 + 10

WhenweFactoraQuadraticExpressionwesimplygointheotherdirection:convertfromthequadraticformbackintoitspolynomialfactors.

𝑥! + 𝑏𝑥 + 𝑐 = 𝑥 + 𝑑 𝑥 + 𝑒 Forexample:

𝑥! + 7𝑥 + 10 = 𝑥 + 5 𝑥 + 2

Because… 𝒙+ 𝒅 𝒙+ 𝒆

= 𝒙𝟐 + 𝒅𝒙+ 𝒆𝒙+ 𝒅𝒆

= 𝒙𝟐 + 𝒃𝒙+ 𝒄Tofactorthiskindofquadratic:𝒙𝟐 + 𝒃𝒙+ 𝒄…Weneedtofindtwonumbers𝒅 and 𝒆,thatwhen:1. multipledtogetherequalthec2. addedtogetherequaltotheb

𝒅 ∙ 𝒆 = 𝒄 𝐚𝐧𝐝 𝒅+ 𝒆 = 𝒃

Example:Factor…𝑥! + 11𝑥 + 30𝒙𝟐 + 𝒃𝒙 + 𝒄

Lookatc:Ifcispositive𝒅 and 𝒆havethesamesign:positivebecause𝒃ispositive.Ifcisnegative𝒅 and 𝒆havedifferentsigns.1)Signs:

2)Listallofthefactorpairsforcandtheirsums:

3)Usethefactorpairthatworksforbothcandb

4)Makethefactors: 𝑥! + 11𝑥 + 30 =

Factorsof Theirsum

1and30 1+30=31

2and15 2+15=17

3and10 3+10=13

5and6 5+6=11

Example:𝑥! + 11𝑥 + 30

1)cispositvethesignswillbethesame.2)Listallofthefactorpairsforc30andtheirsums:3)Usethefactorpairthatworksforbothc30andb11,thatis:5and6

4)Makethefactors: 𝑥! + 11𝑥 + 30 = (𝒙 + 𝟓)(𝒙 + 𝟔)

Factorsof30 Theirsum1and30 1+30=31

2and15 2+15=17

3and10 3+10=13

5and6 5+6=11

Example:𝑥! + 13𝑥 − 30

1)Signs

2)Listallofthefactorpairsforcandtheirsums:

3)Usethefactorpairthatworksforbothcandb:

4)Makethefactors: 𝑥! + 13𝑥 − 30 =

Factorsof Theirsum

1and–30 1–30=–29

2and–15 2–15=–13

3and–10 3–10=–7

5and–6 5–6=–1

–1and30 –1+30=29

–2and15 –2+15=13

–3and10 –3+10=7

–5and6 –5+6=1

Example:𝑥! + 13𝑥 − 30

1)cisnegativethesignswillbedifferent.

2)Listallofthefactorpairsforc–30andtheirsums:

3)Usethefactorpairthatworksforbothc–30andb13:–2and15

4)Plugthemin: 𝑥! + 13𝑥 − 30 = (𝒙 − 𝟐)(𝒙 + 𝟏𝟓)

Factorsof–30 Theirsum

1and–30 1+(–30)=–29

2and–15 2+(–15)=–13

3and–10 3+(–10)=–7

5and–6 5+(–6)=–1

–1and30 –1+30=29

–2and15 –2+15=13

–3and10 –3+10=7

–5and6 –5+6=1