5 2x3+Z*-7x- 30: (x-2)(a** bx* c)

6 f -z*-t 4x*2:(x- r)(xz-2x* a)+ b

7 4f +3e* 5x*2: (x+2)(a*-t bx* c)

g zf + A* _ 8x_ zo:(*_ a)(Bx + c)

9 ax3+ b** cx* d: (x+Z)(x + 3)(r+ a)

l0 ax3 + b* * cx-r d: (4x + t)(Zx - t)(3x + 2)

11 Given thatflx) : 4f - Z* + Zx * 1, find the quotient and remainder when

f(x) is divided by x - 2.

13

13

Given that 5x3 - 6,* +_2x + L

x-2 = A* * Bx * c + #,find A,Band c.

Findthe quotient andremainderwhen xs - 2# - x3 + * + x * 1 is dividedby*+t.

Ed Express each of the following in the form #(a)h+*

(b)#_ffi1

+2, x+2+-'2x* 1

Remainder theorem

TXAMPLE }$

iOLUTION

Find the remainder when 3x3 - Zxz * 4x * 1 is divided by x * 1.

Using long division gives:

3x2-5x*9x*t)zf-2x2*4x*1

-O* + tA-5x2 * 4x

(- sxz - 5x)

9x*1- (e. + e)_

-B

3xz which is the first term in the

quotient.

-5x is the second term of the quotient.

9

3x3x-

')

- 5x'x9x-x-

When 3x3 - 2xz * 4x * 1 is divided by x *the remainder is - B.

We can rewrite this as:

3x3 - 2x2 +_4x + 1 : 3x2 _ 5x + gx*1We can also multiply both sides by (x + 1) and write it as:

3x3 - 2* * 4x t r=(3xz - 5x t 9)(x+ 1) B

1 the quotient is 3xz - 5x * 9 and

B

xll

The remainder theorem

When a polynomial f(x) is divided by a linear expression (x - I), the remainder is (\).

PROOF

When f(*) is divided by * -. f(*) /1/ ^-\ , R

"';q: Q(x) +t-+f(x) - (x - \) Q(x) + R

\, we get a quotient Q(x) and a remainder R.

,(\)-(\-\)Q(}')+R+RThe remainder is/(}').

(Multiplying both sides by (* - \))Substituting x - \ gives:

-.,(\)

Let us use the remainder theorem on Example 11, where we wanted to find theremainder when/(x) : 3x3 - Z* + +x * 1 is divided byx + 1.

Sincewe are dividingby x * I, y : -1 whenx * 1 : 0.

By the remainder theorem, the remainder isl- 1).

Substituting x: - 1 into/(x) gives:

f(-r): 3(- r)3 - 2(- t12 + +1-t; + t- -3-2*4+t- _oo

This is the same answer as when we used long dMsion.

EXAMPLE 13

SOLUTION

Find the remainder when/(x) : 4x3 - x2 + x - 2 is divided by

(a) x-r(b) x+z(c) zx+t

(a) When x - | :0, .tr : 1. By the remainder theorem, when/(x) is divided byx - 1, the remainder is/(t).:.f(1): 4(t)3- (1)2+ (1)

-4-I+t-2-1

(b) When x I 2 : 0, x : -2.8y the remainder theorem, when/(x) is divided byx -f 2 the remainder isf(-2).

f(-2) : 4(-2)3 - (-2)'+ (-2) - 2:-32-4-2-2: -40

(c) When 2x -l | : 0, x : -+. By the remainder theorem, when/(r) is divided br

2.t -.,- l, the remaind.r rr/(+)

l'

llllti

ii

i;iliifr

d,

7A

f(-+): ^(+)'- (+)'* (+)-z-1 - 1- I _ ')2 - 4- 1- z

-, 1-14

.XAMPLE 13

:OLUTION

The remainderwhenflx) : 4f + a* + 2x* 1 is dividedby 3x- 1 is 4. Findtheval:ue of a.

When3x-1:0,x-

Since f(*) : 4x3 * axz

f(+): n(+)' * ,(+)'*

:+*io*?*-|o+fi

Sincef(+)-+

i,*ffi - +

|o:+-fi1- 599"- 27

a- #*,o:!

*. u, the remainder theore m f(+) : 4.

* 2x * 1, substitutin g x - ] Sir.rt

4+) +1

1

: IAMPLE 34

:'JLUTION

The expression 6f - +* + ax * bleaves a remainder of 5 when divided by x - |and a remainder of 1 when divided by x * 1. Find the values of a and b.

Let f(x) : 6x3 - 4x2 * ax * b.

When dividing by x - 1 the remainder is 5.

Now/(l) I :,:, n_*

-2*a*.'.2+a*b-5a*b:3When dividing by x I 1 the remainder is 1.

=f(-1) - 1

f(-L):1,;,1 ;:,;L)z

+ a(-t) + b

.'.-10-a*b-1-a * b - 11 l2l

4(t)2+ a(t) + b

a*bb

tll

T1

Solving the equations simultaneously gives:

2b:L4 tll + l2l

Substituting b - 7 rnto [1] gives:

a*7-3,a:-4Hence, a: -4 and b - 7.

EXAMPLE 15

SOLUTION

Theexpression4x3 - * + ax -lZleavesaremainderof bwhendividedbyx * l aniwhen the same expression is divided by * - 2 the remain der is 2b. Find the values o:a andb.

Letf(x): 4x3 - * * ax -t 2.

\v\4ren x: -L,fl* 1) : 4(-1)3 - (- t)2 + a(-t) + z

--4-l-a-12--u-J

By the remainder theorem,fl -l) : b.

=-a-3:ba*b:-3 tll\,Vhen x: 2,f(2) : 4(2)3 - (2)2 + ae) + 2

:32 * 4'f 2a -12

:2a -l 30

By the remainder theorem,f(2) : 2b.

.'.2a * 30: 2b

:.a*15:b-a-lb:I5 12)

a-tb-atb:-3+15 [1]+[2]+2b: 12

b:6Substituting into [1] gives:

a*6:-3A: -9Hence, a: -9 andb : 6.

v2

Try these 4.2 (a) Find the remainder when 6f - 3x2 + x - 2 is divided by the following.

(i) x-z(ii) r+ 1

(iii) 2x - L

(b) whentheexpression#+ axz - 2x * l isdivided,byx- l theremainderis-lFind the value of a.

(c) Whentheexpressionf - 4* + ax * bisdividedby2x - l theremainderisl.when the same expression is divided by (, - 1) the remainder is 2. Find the

values of a andb.

r+landvalues of

ing.

?k

,bs

ffiKffiffiflXSffi &ffi

1#

By using the remainder theorem, find the remainder when:

(a) a# + l* - 2x* l isdividedbyx - |

(b) 3t' + e* - 7x * 2 is divided byx * 1

(c) t' + A* -x * 1 isdividedby2x -t I

(d) (ax + 2)QP -t x * 2) + 7 is dividedby x - 2

(e) { * 6xz * 2 is divided by x -t 2

(f) +f - z* * 5 is dividedby2x + 3

@) 3# - 4* + * + tisdividedby * - 3

When the expression * - ax * 2is divided by * - 2, the remainder is a.Find a.

The expressi on 5* - 4x * b leaves a remainder of 2 when divided by 2x * l.Find the value of b.

Theexpression3.C * a* + bxl- lleavesaremainderof 2whendividedbyx - 1

and a remainder of 13 when divided by * - 2. Find the values of a and of b.

The expression x3 + p* + qx * 2 leaves a remainder -3 when divided by

x * 1 and a remainder of 54 when divided by * - 2. Find the numerical value

of the remainder when the expression is divided by 2x * l.

Given thatf(x) : 2x3 - 3x2 - 4x * t has the same remainder when divided by

x * a andby * - a, frnd the possible values of a.

Giventhattheremainderwhenfx) :2f - * - zx - l isdMdedbyx - 2'is

twicetheremainderwhendividedby x- 2a,showthat 32a3 - 8az - 8a - 9:0.

The remainder when zx3 - 5* - 4x -l b is divided byx * 2 is twice the

remainder when it is divided by, - 1. Find the value of b.

The sum ofthe remainder when x3 + (\ + 5)x + L is divided by *' 1 and by

x -l 2is 0. Find the value of I'.

The remainder when 3x3 + kxz * 15 is divided by * - 3 is one-third the

remainder when the same expression is divided by 3x - 1. Find the value of k.

when the expression 3x3 + p* + qx -f 2is divided by x2 + 2x * 3,the remain-

der is x * 5. Find the values of p and q.

The expression 8x3 + p* + qx * 2leaves a remainder of 3| when divided by

2x - land a remainder of -1 when divided by x * 1. Find the values ofp and

the value of 4.

When the expression6xs I 4x3 - ax * 2 is dMded byx * 1, the remainder

is 15. Find the numerical value of a.Hence, find the remainder when the

expression is dMded by x - 2.

E2

1&

ainder is 4.

E3

PROOF

The tactor theorem

Th'e f'ac,tor theorern ,

x - \ is a factor of f(8 if and only if f(}.,) - 0.

f(x) , \ . R

g:Q(x)*p\=+ f(x) - (x - I)Q(x) + R (Multiplying both sides by (* - i))Since x - ), is a factor of f(x) =+ F : 0.

When x: l:

.'. f(i) - o

EXAMPLE 1S

SOLUTION

Determine whether or not each of the following is a factor of the expression ,

f+z**2x*t.(a) x-1(b) r+t(c) 3x-2

Letf(x)- x3+2**zx* 1.

{m} Whenx- 1:0,x- 1.

If x - 1 is a factor of f(x), then f(L) - 0

f(L):t3+Z(L),+z(L) +1-I +Z+z+ 1-6Sincefll) * 0, x - 1 is not a factor of f(x)

{b} Whenx* 1:o,x- -lf(-L) - (- 1)3 + 2(-r)2 + 2eD + 1

: -1 + 2 - 2 + 1

- -3 +3

-0Sincefl - 1) - 0 = x * 1 is a factor of f(x).

{e} When3x-2-0,x:

f(?) : (?f *,(tr)' *884

-vI-IrII2793B+24+36+27

?3

4?

__ 95

27

Since f(?) *o

)+l

27

+ 3x - 2 is not a factor of f(x).

74

I. A}tPLE E7

-UTION

Forwhatvalueofkis/(r) :2f - 2* + kx * l exactlydivisiblebyx- 2?

Letf(x) :2f - 2* + kx + l.

Sinceflr) is divisible by * - 2, by the factor theoremflZ) : g.

Substituting intoflr) gives;

f(z) :2(2)3 - 2(2)2 + k(2) + t

:16-8+2k+1:9*2k

f(z): o

+9+2k:02k: -9,---9n-

2

_ {}[PLE affi

-UTION

The polynomial 2x3 l gxz * ax * 3 has a factor x + 3.

{a} Frnd a.

{h} Show that (x + 1) is also a factor and find the third factor.

{m} Let f(x) - 2x3 * 9x2 * ax + 3.

Since x + 3 is a factor of (x),bythe factor theore^f(-3) : 0.

.'. 2(-3)3 + g(-3)2 + a(-3) + 3 - 0

+-54+81 3a*3-0

=3a - 30

+a-10:. f(x) - 2x3 * 9xz * l.ox + 3

{h} If x t 1 is a factor then f(- L) - 0

f(-L) - 2(- 1)3 + e(-r)z + 1o(- 1) + 3

- -2+9 10+3

- -12+L2

-0.'. x + 1 is a factor of f(x).

Now we find the third factor.

Since x + 1 and x + 3 arefactors, then (x + 1)(x + 3) is a factor.

.'. (x+ 1)(r+3) -)et4x* 3isafactoroff(x).

To find the third factor we can divide:

2x*1x2 + 4x t z)zf * 9x2 * tox + 3

- (2x3 + Bx2 + 6x)

x2+ 4x*3-(*' + 4x + 3)

.'. 2x *1 is the third factor. o

Alternative method to find the third factor:

Since f(*) - 2x3 + 9xz + 10x + 3 and x * 1 and x * 3 arcfactors of f(x):

2x3 + 9x2 + 10x * 3 - (x + 1)(x + 3)(cx + d)

To find c and d we can compare coefficients

Coefficients of x3:2 - 1 X 1 X c

.'. c-2Comparing constants:

3-1X 3 X d

3d:3d-1.'. the third factor ts 2x * 1

EXAMPLE XS

SOLUTION

The expression 6x3 + px2 * qx

der of 2 when divided by x - 1.

+ 2 is exactly divisible by 2x - 1 and leaves a ren--

Find the values of p and q.

Let f(x) - 6x3 * pxz *Since2x-lisafactor

f(+) - ,(+)'* o(+)'

-1+L{+la+

-LnP**q++

Since f(+) - o

=L{**q+)-o=pr2qr11 -0P + 2q: -11

Using the remainder theorem,fll) - 2.

Since there is a remainder of 2 when f(*) is divided by * - 1:

f(r) - 6(1)3 + p(1)2 + q(r) + 2

- p + q+ B

qxl2.of f(x)=f(+)

+ ,(+) + 2

2

:0.

(Multiplying by 4)

tll

v6

Since/(l) - 2

+p*q*B:2P + q: 6

Solving simultaneously, and subtracting l2l from [1] gives:

p+2q-p-q--11 (-6)

=q--5Substituting q: -5 into l2l gives:

p--6+sp - -1Hence,p: -l andq: -5.

121

'^! thesm 4.$ (a) Oetermine whether or not each of the following is a factor of the expressionzf-*-3x*t.(i) x- |

(ii) 2x + I

(iii) 3x - 1

(b) fne expression +f + p* - qx - 6 is exactly divisible by 4x * 1 and leaves a

remainder of -20 when divided byx - 1. Find the values of p and q.

Factorising poLynomials and solving equationsA combination of the factor theorem and long division can be used to factorisepolynomials. Descartes' rule of signs can assist in determining whether a polyno-mial has positive or negative roots and can give an idea of how many of each typ.of roots.

(a) fo find the number of positive roots in a polynomial, we count the numberof times the consecutive terms of the function changes sign and then sub-

tract multiples of 2. For example,if f(x) : 4x3 - 3x2 + 2x * 1, then/(x)changes sign two times consecfiively. f(x) has either 2 positive roots or 0positive roots.

(b) to identiff the number of negative roots, count the number of sign changes in

fl-*).The number of sign changes or an even number fewer than this repre-

sents the number of negative roots of the function.

Ifflx) : +f - l* * 2x * l,thenfl-x) : 4(-x)3 - 3(-2s12 + 2(-x) + r

=f(*x): -4f - z* * 2x * tSince there is 1 sign change, there is 1 negative root to the equation.