Factoring Cubic, Quartic, and Quintic Polynomials

The number one rule of factoring is that before anything is done to the polynomial, the terms must be -- - ___..

ordered from greatest to least dewee. Beyond that, there is a series of things to check for to break thepolynomial down into prime factors.

1. "Undistribute" the greatest common factoiliom .. each tenn, if possible. If the leadingcoefficient i-s negatiYe, factor out the opposite of the greatest common factor.

2. If.the polynomial is a.binomial, check.to see · it is a difference of two perfectsguares.

· 0-fl._ b � : ( �-\-'o )(� _ b)

2.x3 - 32x 12x - 3x3

� ')( ( � '2- - , 6) - 3 � 1 � , 'lxl.)( (x-+41\ ;(��) -3x( � 1-_ �)

-?>x (�-\-�)�-�

2x4 - 32

.'2- (� 4 -Ho):i.(���� c��-t.t) 2(1.1.."t �) �+ 2.) �- �

3. lf the polynomial is a binomial, check to see if it is a sum or difference of two perfectcu bes. ' • 'It '1, '=- 4, 12..,;., :2. \ '-

()..� wt b J: (Q.,1 b")( a... '1.-a.,\, t b:J.J a.;-b1� (o..- b )(a.�.+al,-+ b:r)

x3 -27 2x4 + 16x -3x4 -192x

(�- ?>)(�::i.+3x +�) �x ( 'll. '!.� �) -!x(� �""" <o4) � (�..\-�)(��-:«� -3x.(�+4\ )(�"-4'xrtu.)

4. lLthe polynomial is a trinomia!, check to see if it :will factor as two differentbinomials.

2x4 - 1 3x2 + 20 ·

(Q-,.."- 5) (x'J. - 't) (-x 2._ \ ) (x � � 5 )(2.-,..

'l.- s) (�+2.)�-�, (.�-t-l)(x-i)(x"--+&S)

2x5 -10.x3 + 8x

�x( � 'i_ 'S� �-+ 41-) ':2 '>< ·()(a - \ ) (x� - 4: �x (x·H )(�-()(x-+t()-.:2)

5. If the polyno1nial has an even a1nount of tenns, try to factor by grouping.

Factoring by Grouping:

Often, conventional methods of factoring are not useful. In cases where the polynomial contains an even number of terms, a technique called grouping can often be used.

I. Order the terms from greatest to least degree.

2. From the first two terms, factor out a GCF.From the second two terms, factor out a GCF.

3. Step 2 should produce two different groups thatnow share a GCF. Factor that GCF out of eachgroup and write what remains from the two groupsin a separate factor.

4. If either factor from step 3. can be further factored,then do so.

I. Order the tenns from greatest to least degree.

2. From the first two terms, factor out a GCF.From the �econd two terms, factor out a GCF.

3. Step 2 should produce two different groups thatnow share a GCF. Factor that GCF out of eachgroup and write what remains from the two groupsin a separate factor.

4. If either factor from step 3 can be further factored,then do so.

Factor each of the following polynomials by grouping.

x3 + 3x1-x-3 \ J ' '

X 'i( ')( -1- ?>) - \ ( � -+ �

()(-It�)(� 2._ i) (�-\7 t C�-+Dtx-\)

2x3 + 3x2 8x - 12

2x3 +x2- 8x-9

4-��c�-'-")-,(�-4)

('I..- 4) ( � 'k-i-,)("-. - 4) (�� � \)(2.� -�

Factor each of the following polynomial functions whose equations and graphs are provided. Then, set each factor equal to zero and find the values of x.

F(x) = 3x3 + 12.x2 + 12x 4 0

t=(-.c):: 3� ( � �� 4x-+ 4)

= 3>t (x -+2.)(x+2)

G(x) = x4 - 5.x2 + 4

......... , .......... 1 '

· ·······+ ..... , . ...jl. ............. , •..••.........• , .•.............

.... , ....... -4 ................... , ............... , .............. .

What is they- intercept of the graph? (0 .. o) What is the constant tenn in the equation? Q

··,···········,·····-5· l \

{;..()t) ': (')( 2-_ 'i )( � 'l-_ \ )

G,.("K-)-:... (X.� �)Q(-!>-)(X�\)(_x. \)

x.-\ 0

)( :..\

Hi(x) = 2x3 + 5.x2 - 2x - 5 \ " ,I

\.\C,.) : 'X.�c �to\- cs)-\ ( � "T cs)

: ( 2x-+ s)(x �- l) - (2� 9': s) C � � U(><. -I)

��+s� o 2.)l-::.-�

x:.-%.

)(-t\ � 0

y .. ::. -\ �-\:0

x�,

........... L ..... 3. . .... . : ........... : ........... : ........... :

What is they - intercept of the graph? ( 0, � What is the constant tenn in the equation? 1

: : :

r···+····+·· ·

i 3 4 3 : i i : 1 1 i

.... +····· .. +·······!·······+·······{ . .

1 : ; : :. .

What is they - intercept of the graph? �

What is the constant tem1 in the equation? -S-

P(x) = 2x4 - 5.x2 + 3

�) 7- (i�2. - ?>;(x�- ·,)

� c�:2.-�)(�+\ )(x-\) �

�x.-?> ::. 0

2.�1..= 3

Y:J1 �:: ! \.'115

x-t\: o

�=-\

r:::::::::::::r:. ·::::::::r:::::::: ::::::::::::r::::::t:r:::::::::::::i ! I i I

1 ;

! ..... ......... ! ..... ...... "i" ........ 3- ..... ....... 1

.... ... ...... , ........ ...... !

-� f :l i 1 L .............. L .............. L ...... wl· ........................................... ..

What is they- intercept of the graph? (o' �)

What is the constant tenn in the equation? .3,.

Date -------Period ---

Factoring Practice

Completely factor each of the following polynomials.

1. 12.x3 - 14.x2 - 6x 2. 6.x3 - 3x5 + 24x 3. -4.x2- 14.x3 - 6x4

�x( "x?-- 7x - 3) - 3x 5 -r tcx -s -td-1K -b)('-1- l�)<..3 -Lf}(. -;i

�x: (_3x.-\- l°) (� --3) -3-,<()( L.\ -lx.1 -Q) - ix:1 ( 3 x:1

-t 1 � -+ i)

-3x (x�-\- �)(')l,_�-'i-) �.2,.)(2( 3x. -t \ )(� -t :l') l

-3x (X:;- ;))(x. ;-.).)(x -.;}.)

4. x3 -3.x2 - 9x + 27 5. 3.x2 - 2.x3 + 8x - 12 6. -2.x3+18x

x�(x-?>)- ,(x-3) - �}£. 3+3x :>.--+ 8x -1 � -.:l. x ( x?· - '1)

(�-3) (x >-- Cf) -x?.�'/..-3)+��-�) -��cx+3)(x-3)(� -3)(- '1-

:2.

-t 4 J(x-3)(x-t 3')(x-3) -{�.:.. 3) (x. -\-)._) (x-;l..)

( � -+sXx - 3J'J-

7. 2x4 - .x2 - 15 8. x4 -.x2 - 12 9. 6x5 + 9.x3 - 27x

(�-;...1 ;- 5)(_xi - 3) ('I.. :1. -t � )(xi - 1 J 3. X ( 2$ 't -t ?:,';... '2. - 9){_y._')-+-3)( X � �X - :)_) 3x (2.)(. i_ 3)(x :).t- 3)

Factor each of the po ynomial functions. Then, identify the zeros of the function. Show your work. 10.j{x)=5.x3-20x. 11. g(x)=3x3 -3.x2- 18x 12. h(x)=-10.x3+26x2+l2x

�)� '5x.(�i - 4-) 'jv-) == 3x(� 2.. _ x - b) h6'):: -.l.x.( s�1-\3)<..- �) �)� sx(..x+i)(.x-c:lJ �)=- 3'AC:,..-3)()(..·\<1.) \,(2'.)= -d-x (s--�t-� --3)sx�o X-t:L=-D x-�-=c 3x=-0 x-3:::.-o x t-.;l=o -�x.. =o 5">,-+�-=-c ')( - 6-::.(:) X:::.. D X � -� x:=· � ')( ::.a )(:=-3 X ::.-,;t �::::. 0 X= -i 'X :� 3

13. p(x) = x3 + 2x2-4x-8

{)()() :;. "'?.

& -t :).)-L\ (x-t ). 14. q(x) = 3x3 + 5.x2-3x-5

Ci)'.:: )2{3x'*5)-,(3x+5 15. r(x) = x4 - 10.x2 + 9

r&) � C x?--'t)(x 'l.._ 1)

�-=- (x+�(x 2- \.\)�::. (�kJ..)(�-t-.?.-)(x-::l

[j_) = (3X-+ s)(x 1 - ') r6<)::: l� -ttQ(-3)(_'t.--t\�-) =- G)(+S)C.X.-+\X',l-0

�:t3 ::.o X. -3-:::-0

3x-\ 5::: D x.-+ \ :::.0 X-\-=-0 x_::..-3 )(:::. 3 ,><..:::.-\

x� -� x�-\ x=-\

16. m(x)=x4-2x3 -I5.x2 17. g(x)=4x3+16.x2+16x 18. h(x)=2x3 +3.x2-8x-12

rr-1.J..) = �:l

e x.1.-�)(_-\s j(i() = '-t-�(x1 -t- Y-1' * v "<0 = ?<,:l.('.L�-t� -�Gx�3

frQ0 ::: )'..1

( x. -'5 )0,--I- 3) SJ-) = IJ-x.( x-.. ::>-)()..--\ d_:) f{,i)=- � t-t{_ x?--ti) y .. ,.')---=-0 X -5

-::.0 � \-3={: 1i-'l----:.-O X-t l. =0 \-\(j<.) :=. (:2><. � 3)( X.-+ .).x_ x-,�X:=-:, X =--3

)l..=-D

Pictured below is the graph of a polynomial function,.f{x). Use the graph to answer the questions that follow. 19. Identify the

zeros of .f{x).! ..... ) ....... ! ........ ! ..... ...1 .. : ...... ! ..... t ' . ..!. .... ...! ........ ! : : : : : : : : : : � ........ , ........ f········!········+···l· ·····+·······1 ·······f ········l········i 1 ........ 1 ........ 1 ....... H .. l· ...... l········' ....... 1 ........ 1 ........ 1

1Jil':"� JtJJJ i_J_tvll-�B?J LL!

22. Rewrite the factored equation in standard fonn.

�)'.::. �-+.1.-)(.x-d-)(x-t,)

W)::. ( x?--u/)[ X. + ,)

-f{�J-= 't. 3 + x.'2..

-4 � -L\

20. Based on the graph, what are the coordinatesof they- intercept?

21. Write the equation ofj{x) in factored form.

23. What is the constant term in the standard fonnequation of j{x)? State the connection thatthis term in the equation has with the graph ofj{x).

� ��� -\-� (. 5 --'\ . . IL .' II , d . - \..\ � - _J -� _-f..,.,. �°"- \� .� <....J- �- - -

� � �-� v-=\-� 1) � �> (OJ-4J-