fl*
r
Factoring by grouping is a method which enables you to factor polynomials by grouping the
terms which contain a common factor
You then remove the common factor in each of the groupings:
ac + ad * bc * bd. : lac + adl + (bc + bdl
a(c*dl+b(c+dl
: (c *
dl(a
+ b\'
Ex.: Factor the following expression using factoring by grouping.
p(x) : g*
lZ*y? * 6xy - 8y3 <- Group the terms containing
-
:3xl3x -
:
6.
(3x
-
4yz)
4y?l{3x
+ 7yl3x +
1)
a
common factor.
4y2l€ Remove the common factor in each grouping'
+ Remove the common factor a 2nd time.
Factor the following polynomials.
a) xz +5xy-t3x* rsy (X+S)(X+sU), U1 zxz +3xv - 70x_ 15v
c) 6az - 15a * Zab - sb (3 a+uXa" -s)61 6xz - 8x - 9r4t + lZv
e) Toxy + 2x -t tsy + s (axXXsU rD f) x3 - xz * x - |
7.
Factor the following polynomials.
a)
.
_8.
xx
Zx\t + 3xz + lQy + 15
ts$? + 3siyz * e* -
71
b)
c) 7x3 + 4*) * Zxz - 4xy
d) 3*3y - 9x3 + 6*) * 18x2 3x e
e) 3Oxay - l0x3y2 * 15x3y - 5*\'
f) Zxa * zx3 + 6x2 - 6x
Factor the following polynomials' [rrrc a ?'u*?s)
a) %__ary+bry=W+ry_-_g
- : " y b) 6ax - 3av -t 10bx - 5bY - 4x +2Y
c) a3 - Zab + ac' *-db + 7L/ -.k? + ozq-&, +-C
(ax -bj)(hx - nJ)
d) oU, * y'l -
Wazt
a'nd
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Factoring a Polynomial 1l
Aeruvnrv 3
oitterence of squales
The following remarkable identity enables you to factor
a
\*
difference of squares.
e)rPand
:
(a+blb-bl
^
L_/
az
-bz
factor
a) Factor the following differences of squares.
1. x2 -25
2. 4xz - 9y2
)x'-
/
4. -x2
+
9
t
1)z
3.
5.
(3x
- 4*z
b) The rectangle on the right has an afea of A
' possible dirirension, of tirit rectangle?
'
:
16x2
-
9-
What could be the
fI A : .- - eI I
16x2
llfrEnEtscr drsnunnrs
' ''''' ',' ,..-'--.-..
"""'{rIfFEnEilCE
Of'SOUARE
"
?
. "...= _,,,,.,*;;-,,,,-.--,""'
o A difference of squares is an algebraic expression of the form az - bz'
o Every difference of squares is factorable. You simply need to apply the remarkable identity:
a2-b2:(a*b)(a-bl
Ex.: Factor:
: (3, + blbx - }yl + Apply the remarkable identity'
. llx+ 1)2 - 36 : {?x * 7lz - 6z
: llTx + 1) + 6l[l7x + l) - 6]
.
: i2* + 7)(2x -'i\
(3x+5)2
+ (7x-t l)][(3x+5) -(2x+1)]
-llx+t)2:[(3x+5)
: (5x 6)(x + 4)
+
A sum of squares is not factorable.
9. Factor the following differences of squares.
=b a) xz - 25 (xts)tv -5)
-? c) 4ex2 - soyz [7x +b]) t1x -69
e) loo-r' (to+-)(ro--)
g) x2-3 [X*
i) t6x2 -f, (,rx - i)(rl -i)
'18
Chqpter
I
Algebraic expressions
-g
+t)({v -S)
d) 36*!i -zsizo &'a+ srl)/ r, a-'r,
(1x
b)
t6xz
f)
ft + 4.+>(+',9
h)
75
xryo
-
i) 16
-
lzu
9
'v \23
t-
"r{
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