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VL. Zz. dence. Identify the graph ofthe function f(x)= (x 2)(x=1)7(x+1). \ L. p(x)=(x+3)(x 1)",as p(x) approaches the x-intercept at x = 1.

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Pre-Calc 40s Past Achievement Test Questions on Polynomials

June 2015

Question 8

1 mark

Explain howthe value of n affects the behaviour of the graph of the polynomial function

p(x)=(x+3)(x—1)",as p(x) approaches the x-intercept at x = 1.

2 y= \ ane Qc w tll Cress Drroyht =|

°

=

\

L

. \¢< nN is odd and G cea Aner re qa ue

wll level ff as tk crosses He xK-Gus

\f£ WwW is even the Amen enki U Dienek elf tLe

K Mtetpr

Question 19

1 mark

Identify the graph ofthe function f(x) = —(x—2)(x=1)7(x+1).

qe

dence

a) F : Negehor leading « [i A

.

Co exc iz i

@)

=

VL.

Zz

po

A ¥

(2)

Question 28

3 marks

Determine the equation of the polynomial function represented bythe graph.

A

ad = 4

\i

2

=A

2 *

p(s)=_ dC x43)%-2)

Question 39

a) 2 marks b)1mark c) 1 mark

Christine dives off a diving board.

* ° 4 4 . * 3 3 . e *

Herdive is modelled bythe function A(t) = 1° - 317 -*+3, where h is her height in metres,

relative to the water surface and f is the time in secondsafter diving off the diving board. a) Given that (1+ 1) is a factor for the function (1), determinethe other factors.

“(|

fs => -b

3

The obhec factons

«4 4 ~3 ace

( €- 3) and

C¥-\)

=

(&+*\) (4t 4k? )

= (ee) (b-3) (EY)

week)

(3)

b) Sketch the graph ofthe function A(t) for the timeinterval t = 0 tor = 3.

h

A

>

oO

/Y” [! GQ

c) Determine how manyseconds Christine is underwater.

a Seconds

January 2015

Prom \su —2

ZSEC-Question 9

2 marks

Is (x ~ 2) a factor of the polynomial p(x) = =x4 —3x3+11n7 +3x-10?

Justify your response.

e Cab ee Ss te Kee)” + 3(2) -—

= = V& =3 Le\+ wl) 4 & ~O

= =U - U4 ed -Y

~

oO

(a-e%) tha Qaeree.

Question A S MCE. C2) =o we Ca erent Traorem stales 1 mark ot Lx)

Explain howthe end behaviours of the graphs ofpolynomial functions with an even degree and with an odd degree are different.

Ey en deaee \Polunom tals + ep We daleo QE ead

or

aosen

Me

QW ns

VUdule gad A egece pol yao ats

evtier so doun rho

Q wt

and ve we QL

Oo

—_—_——

(4)

Question 24 1 mark Whatis the degree ofthe polynomial function represented bythe graph below?

a) 2 b) 3 c) 4

d) 5

, & &

Question 27

x 2 marks . . . 2 og

Determineall of the zeroes of the function p(x)= x3 - 5x7 -2xt 24, given one of the factors

of p(x) is (x—3).

3

L

-S

-2

24

4 =G& =24

Question 33

| | -2

8

©

os (v3) (x an-k)

O= CxK-32)C%- 4) (xeZ)

i Sy ty =

3 marks

Sketch the graph of y = -2(x-1)(x-3)(x+1).

*

y= - zC-v\)(3) cy)

yak:

(5)

June 2014

Question 8

a) i mark b) 1 mark

Let PLK) =

a) Determine the remainder when x* -— 3x7 +1 is divided by x+2.

Pl-2) = €2 yt 3-274

=

(b—3 (4) +\

=

Ss

Question 16

2 marks

2.

KSA x —-x—-4

K+

Divide (x> — 5x —4) by (x +1).

Question 28

1 mark

A sheet ofpaper 12 cm long and 8 cmwideis used to make a box withnolid. Equal squares of

side length x cm are cut fromeach ofthe comers andthe sides are folded up to makethe box.

xf T T cree re t I I | ! }8 cm I I

late fe we zo J

I I

Which of the following expresses the volumeof the box?

a) V(x)=x(12+x)(8+ x)

b) V(x) =x(12-x)(8— x)

c) V(x) =x(12 + 2x)(8 + 2x)

(a)) V(x) = x(12 — 2x)(8 — 2x)

(6)

Question 42

2 marks

Write the equation for f(x) that satisfies all of the following conditions:

¢ f(x) isa polynomial function of degree 4

¢ f(x) has zero at 2 with a multiplicity of 3

3

* f(x) has a zero at —5

.

Wao C™« xS)) («- =)

¢ f(x) has ay-intercept of 80

w= als \(-e)

—Q = G4

4 >-a Cxs) ix-2°

Question 45

3 marks

Sketch the graph of f(x) =(x—- 1)7( x+2 ;

‘ fe

Label the x-intercepts and the y-intercept.

=

5

i

Sc@)y= C1) (+2)

(7)

January 2014

Question 20 1 mark

- * be

LY

¥

Given the above graph of a polynomial function, which one of the following statements can be true?

a) The function has a degree of 4 with a positive leading coefficient. b) The function has a degree of 4 with a negative leading coefficient. c) The function has a degree of 3 with a positive leading coefficient. (@)The function has a degree of 3 with a negative leading coefficient.

Question 21

1 mark

Given that (x+ 3) is a factor of polynomial P(x), which ofthe followingis true?

a) P(-3)=0

b) P(0)=-3

c) P(0)=3

d) P(3)=0

Question 26

2 marks

,

4

P(+2)=0

One ofthe factors of P(x) = x" — kx” — 7x +10 is (x-2).

Findthe valueof k.

O= ry = Kary’ -4Fz)4tO

O= *£% “HK —-Ws4rL0

~ My. = ~-4K

Question 35

,

= te

1 mark

When P(x) is divided by x —3, it has a quotient of 2x7 +x —6 anda remainderof4.

Determine P(x).

ex) ~—

axrexrlo +4

x<3

X~-3

(8)

Question 40 4 marks

2 z 4 ‘

Sketch the graph of y= ta? —5e43 given that one ofthe x-intercepts is 1. Identify the x-intercepts and y-intercept.

a June 2013

Question 24

45 Cet) (x4 rx-3)

4 = (x ~\) Cx 3x)

+

4

_ (3)

4

)

=

3

4 >

1 mark

Whatis the degree of the polynomial represented below?

fo

a) y

A b) 3

(9)

-Question 29

\y_

3 marks

a

Sketch the graphof:

BSC),

f(x) =(2 -—x)(x + 3)(x + 1°

(10)

Question 31

1 mark

5

The graph belowrepresents the equation v= ax? + 6x" + 5x -10.

¥

A

V\

Vv

What must be true about the value of a? Explain your reasoning.

O

+s

Rename »

Since tore

exc

Vy erwvi oF Ls ve ino Q w a~d dosa cable QW

Question 35

2 marks

Giventhat (x — 1) is one ofthe factors, express x — 57x 456 asa productoffactors.

-st

Sb

&

)

ij} =

Cx 1) Cx *e x - SG

Pr.

\ -Se 0

Cremt) C xe OF)

Question 44

2 marks

(4

4.3. 4.2

.-Is (x — 3) a factor of x" —x° — 3x7 +x-1?

Justify your answer.

4

ar

2

_

e Ca) =

3'_ 33 _3(3)~ 3!

=«at -ages-!

=-2Q1

‘atiee:

P(3) # O

(«~3) ds

not we FACroe

(11)

January 2013

Question 27

ad

Sketch the graph of y =(x+1)(x- 2)? (x+5).

Identify the v-intercepts and 1-intercept.

Lo

y-intercepts:__— Ss ] Tit,

y-intercept:

a* Cv Ne z)* is)

{~~ =

11

3 marks

(12)

Question 31

5 marks

A box inthe shape of a rectangular prismhasside lengths x. + 2. andy +10.

Write a function. V(x). to express the volumeof the box in terms of x.

-

.

3

Find all possible values of x. given that the volume of the box is 96 cm”.

State the dimensions of the box.

V Cx) =

*(x*2) ( x+t0)

PLix)= (x-2) (x *e14% +48)

Vo = an

é ©

Pix) = Cree) ( +8) (eb)

= x x + IBTxkto

3

. 2

oO = (ye - BY LeBIC x +b)

O = * ¥\ITK *+a0% =e

|

4 PLxK)= %? eIUK* FaA0x -TS

w = 2 , ~f# \ Me

(2) £0

2. \ i~ wo - %&

2

2%

AL

} § Wo 4S

0

Question 40

1 mark

Giventhat h(x) = 2x7 + Sx —3 and that h(x) = f(x)« g(x). determine f(x) and g(x).

hdx)= C(ax-t) (x4 3)

prise

Sexy = @x-3)

Cuxy= Aax**Sx-5

.

Acx) - Cx 3) Cx) = A

Question 47

+ mark

If p(x)= x? —12x +1. determine the remainder when p(x) is divided by (x+2).

P\X

P

elt) = C2 ster) +1

= -ZS27 +a +] (Lemtadat

= Sel

|

a SB,

<< = he

References

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