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4
3
2
1
Outside the high school program
/ Algebra: second degree reduced to first degree
25 exercises (including 22 corrected)
a
Second degree: canonical form
(5 exercices)
Algebraic expressions
(3 exercices)
Equations
(5 exercices)
Canonical form and extremums
(5 exercices)
Canonical form and direction of variations
(3 exercices)
8
cm
1.
Se
c
ond
de
gr
e
e
:
c
anonic
al
form
E.600
W
e
will
fo
cus
on
second-degree
p
olynomi-
als
:
functions
of
the
form
x
↦→
a
·
x
2
+
b
·
x
+
c
with
a
,
b
,
and
c
b
eing
fixed
real
n
um
b
ers.
W
e
accept
the
follo
wing
prop
osi-
tion
:
Prop
osition
:
An
y
second-degree
p
olynomial
function
can
b
e
written
in
the
form
:
¸
·
(
x
+
˛
)
2
+
‚
This
form
is
called
the
canonical
form
.
First
metho
d
1
a
Expand
eac
h
of
the
remark
able
iden
tities
b
elo
w
:
a
(2
x
+
1)
2
b
(3
−
2
x
)
2
c
(3
x
+
7)
2
d
(
−
4
x
+
1)
2
b
Deduce
the
canonical
form
of
eac
h
of
these
second-
degree
p
olynomials
:
a
9
x
2
+
42
x
−
5
b
4
x
2
−
12
x
+
16
c
4
x
2
+
4
x
−
8
d
16
x
2
−
8
x
+
7
2
Using
this
metho
d,
deduce
the
reduced
form
of
25
x
2
+
20
x
+7
.
Second
metho
d
Consider
the
equalit
y
:
a
·
x
2
+
b
·
x
+
c
=
¸
·
x
+
˛
2
+
‚
1
Sho
w
that
the
n
um
b
ers
¸
,
˛
,
and
‚
m
ust
satisfy
the
fol-
lo
wing
system
:
¸
=
a
2
·
¸
·
˛
=
b
‚
+
¸
·
˛
2
=
c
W
e
assume
that
for
an
y
v
alue
of
the
triplet
(
a
;
b
;
c
)
,
the
previous
system
has
a
unique
solution.
2
F
or
eac
h
of
the
follo
wing
functions,
determine
the
v
alues
of
¸
,
˛
and
‚
:
a
:
x
↦−
→
x
2
+
8
x
−
9
;
b
:
x
↦−
→
2
x
2
+
12
x
+
4
c
:
x
↦−
→
6
x
2
+
6
x
+
7
;
d
:
x
↦−
→
−
2
x
2
+
3
x
−
4
Bon
us
question
1
Sho
w
that
for
an
y
n
um
b
ers
a
,
b
,
and
c
suc
h
that
a
=
0
,
w
e
ha
v
e
:
a
·
x
2
+
b
·
x
+
c
=
a
·
x
+
b
2
a
2
−
b
2
−
4
ac
4
a
2
What
can
y
ou
sa
y
ab
out
the
equation
a
·
x
2
+
b
·
x
+
c
=
0
si
b
2
−
4
ac
<
0
3
W
e
accept
the
fact
that
:
a
·
x
2
+
b
·
x
+
c
=
a
·
x
+
b
+
b
2
−
4
ac
2
a
x
+
b
−
b
2
−
4
ac
2
a
Giv
e
the
t
w
o
solutions
to
the
equation
2
x
2
+4
x
+1
=
0
E.2009
W
e
are
in
terested
in
equations
of
the
form
:
(
E
)
:
(
x
+
˛
)
2
−
‚
=
0
1
a
F
actor
:
(
x
+
2)
2
−
9
.
b
Solv
e
the
follo
wing
equation
:
(
x
+2)
2
−
9
=
0
.
2
F
or
eac
h
of
the
follo
wing
equations,
giv
e
the
set
of
solu-
tions
:
a
(
x
−
7)
2
−
3
=
0
b
(
x
+
1)
2
+
2
=
0
c
(
x
−
8)
2
=
0
3
Whic
h
condition
m
ust
‚
v
erify
for
:
a
(
E
)
admits
t
w
o
solutions
;
b
(
E
)
admits
a
single
solution
;
c
(
E
)
admits
no
solution.
E.6035
In
a
cylinder,
whose
base
has
a
radius
of
8
cm
,
a
ball
is
placed.
W
e
need
to
p
our
104
cm
3
of
oil
in
to
the
cylin-
der
to
just
co
v
er
the
ball.
Determine
the
radius
of
the
ball.
E.4454
Establish
the
follo
wing
equalities
:
a
(
x
+
2)
2
−
4
=
x
2
+
4
x
b
(
x
−
9)
2
+
11
=
x
2
−
18
x
+
92
d
(
x
+
3)
2
−
12
=
x
2
+
6
x
−
3
e
2(
x
+
1)
2
+
4
=
2
x
2
+
4
x
+
6
E.4457
F
or
eac
h
of
the
follo
wing
equalities,
giv
e
the
v
alue
of
¸
(without
justific
ation)
,
then
v
erify
the
prop
osed
equalit
y
:
a
(
x
+
2)
2
−
5
=
x
2
+
¸
·
x
−
1
b
(
x
−
3)
2
+
7
=
x
2
−
6
x
+
¸
c
2(
x
+
1)
2
−
9
=
¸
·
x
2
+
4
x
−
7
d
7(
x
−
2)
2
+
1
=
7
x
2
+
¸
·
x
+
29
e
(
x
−
5)
2
+
¸
=
x
2
−
10
x
+
10
f
3(
x
+
¸
)
2
+
4
=
3
x
2
+
48
x
+
196
2.
A
lgebr
aic
expr
essions
h
ttps:/
/c
hingmath.fr
sacados/600
sacados/2009
sacados/6035
8
cm
chapExoCorrec/4454
sacados/4454
chapExoCorrec/4457
sacados/4457
E.435
Consider
the
follo
wing
algebraic
ex-
pression
:
E
=
(
x
+
1)(2
x
−
1)
1
Based
on
the
reduced
expanded
form
of
the
expression
E
,
answ
er
the
follo
wing
questions
:
a
Find
the
co
efficien
t
of
the
term
x
2
.
b
Find
the
n
umerical
term.
2
Justify
that
the
expression
E
is
not
equal
to
either
of
the
follo
wing
t
w
o
expressions
:
F
=
4
x
2
+
x
−
1
;
G
=
2(
x
+
1)
2
+
1
E.441
Consider
the
function
f
defined
b
y
:
f
(
x
)
=
−
6
x
2
+
10
x
−
4
1
Establish
the
follo
wing
equalities
:
f
(
x
)
=
2(
x
−
1)(2
−
3
x
)
=
−
6
x
−
5
6
2
+
1
6
2
Calculate
the
image
of
the
n
um
b
ers
b
elo
w
b
y
the
func-
tion
f
a
0
b
2
3
c
5
6
E.3001
Answ
er
the
follo
wing
questions
with-
out
justification
:
1
Consider
the
p
olynomial
P
=
2
·
(2
x
−
1)(3
−
x
)(
x
+2)
:
a
Giv
e,
without
justification,
the
co
efficien
t
of
the
de-
gree
term
3
of
the
p
olynomial
P
and
the
v
alue
of
its
n
umerical
term.
b
Whic
h
of
the
p
olynomials
b
elo
w
is
the
expanded
and
reduced
form
of
the
p
olynomial
P
?
−
4
x
3
+
6
x
2
+
22
x
+
12
4
x
3
+
6
x
2
+
22
x
−
12
−
4
x
3
+
6
x
2
+
22
x
−
12
4
x
3
+
6
x
2
+
22
x
+
12
2
Determine
the
v
alue
of
a
,
a
real
n
um
b
er,
v
erifying
the
follo
wing
equalit
y
:
(2
x
+
1)(3
x
2
+
a
·
x
+
1)
=
6
x
3
−
7
x
2
−
3
x
+
1
3.
Equations
E.4404
Solv
e
the
follo
wing
inequations
:
a
(2
x
+
1)(
x
+
2)
<
0
b
(3
−
x
)(2
x
+
1)
0
c
(5
x
+
1)(
x
−
2)
>
(3
+
x
)(
x
−
2)
d
(
x
+
3)(5
−
x
)
2(
x
+
3)
E.4453
Solv
e
the
follo
wing
inequations
:
a
(
x
+
2)(
x
−
3)
>
0
b
(
x
+
2)(5
−
3
x
)
+
(
x
+
2)(
x
−
2)
0
c
(2
x
+
1)(3
x
−
1)
+
(6
x
+
3)(3
x
−
1)
0
d
(2
−
3
x
)(
x
−
5)
+
2(10
−
2
x
)
<
0
E.4452
Complete
the
table
of
signs
for
eac
h
of
the
expressions
E
:
1
x
−∞
−
3
−
1
2
+
∞
2
x
+
1
0
3
+
x
0
E
=(2
x
+1)(3+
x
)
0
0
2
x
−∞
3
4
2
+
∞
x
−
2
4
x
−
3
E
=(
x
−
2)(4
x
−
3)
3
x
−∞
+
∞
2
+
x
2
−
x
E
=
2+
x
2
−
x
E.9424
Solv
e
the
follo
wing
inequalities
:
a
3
−
2
x
3
x
+
1
>
0
b
3
x
+
1
x
+
1
<
x
+
1
4
x
+
2
h
ttps:/
/c
hingmath.fr
chapExoCorrec/435
sacados/435
chapExoCorrec/441
sacados/441
chapExoCorrec/3001
sacados/3001
chapExoCorrec/4404
sacados/4404
chapExoCorrec/4453
sacados/4453
chapExoCorrec/4452
sacados/4452
chapExoCorrec/9424
sacados/9424
E.4409
Let
(
E
)
b
e
the
second-degree
p
oly-
nomial
:
(
E
)
:
x
2
+4
x
−
12
1
Determine
the
canonical
form
of
a
Justify
that
the
p
olynomial
(
E
)
admits
as
canonical
form
:
(
x
+2)
2
−
16
.
b
Justify
that
the
p
olynomial
(
E
)
admits
for
minim
um
−
16
.
2
a
Deduce
from
question
1
a
the
factorized
form
of
the
p
olynomial
(
E
)
.
b
Deduce
the
t
w
o
ro
ots
of
the
p
olynomial
(
E
)
.
3
Dra
w
up
the
table
of
v
ariations
of
the
function
f
defined
b
y
:
f
(
x
)
=
x
2
+
4
x
−
12
4.
Canonic
al
form
and
extr
emums
E.4407
Consider
the
expression
−
2
x
2
+8
x
+
1
:
1
Establish
the
follo
wing
equalit
y
:
−
2
x
2
+
8
x
+
1
=
−
2(
x
−
2)
2
+
9
2
Deduce
that
this
expression
reac
hes
its
maxim
um
at
x
=
2
.
What
is
its
maxim
um
v
alue?
E.8516
1
Determine
the
canonical
form
of
the
expression
:
x
2
−
10
x
−
2
2
Justify
that
this
expression
admits
as
minim
um
v
alue
−
27
and
that
this
v
alue
is
reac
hed
in
5
.
E.8517
Consider
the
p
olynomial
x
2
−
6
x
+12
:
1
Determine
the
canonical
form
of
this
p
olynomial.
2
Justify
that
this
p
olynomial
is
strictly
p
ositiv
e
for
an
y
v
alue
of
x
.
E.4481
Consider
the
p
olynomial
6
x
2
−
4
x
+2
.
1
Determine
the
canonical
form
of
this
p
olynomial.
2
Justify
that
4
3
is
the
minim
um
v
alue
tak
en
b
y
this
p
oly-
nomial
on
R
.
E.8518
Consider
the
function
f
defined
on
R
b
y
:
f
(
x
)
=
3
x
2
+
a
·
x
+
b
where
a;b
∈
R
Determine
the
v
alues
of
a
and
b
so
that
the
function
f
reac
hes
its
minim
um
v
alue
2
for
x
=
3
.
5.
Canonic
al
form
and
dir
e
ction
of
variations
E.7252
Consider
the
function
f
defined
on
R
b
y
the
relation
:
f
(
x
)
=
x
2
+
6
x
+
2
1
Justify
that
the
function
f
has
the
canonical
form
:
f
(
x
)
=
x
+
3
2
−
7
2
Establish
the
decrease
of
the
function
f
on
the
in
terv
al
−∞
;
−
3
.
Reminders
:
w
e
will
use
the
follo
wing
t
w
o
prop
erties
:
Since
the
square
function
is
decreasing
on
R
−
,
w
e
ha
v
e
:
a
<
b
<
0
=
⇒
a
2
>
b
2
whic
h
translates
to
the
sen
tence
:
ˇ
t
w
o
negativ
e
n
um-
b
ers
and
their
squares
are
compared
in
rev
erse
order.
ı
The
square
function
is
increasing
on
R
+
,
w
e
ha
v
e
:
0
<
a
<
b
=
⇒
a
2
<
b
2
whic
h
translates
to
the
sen
tence
:
ˇ
t
w
o
p
ositiv
e
n
um
b
ers
and
their
squares
are
compared
in
the
same
order.
ı
E.7253
Consider
the
function
f
defined
on
R
b
y
the
relation
:
f
(
x
)
=
−
2
x
2
+
6
x
+
1
1
Sho
w
that
the
function
f
admits
as
canonical
form
:
f
(
x
)
=
−
2
·
x
−
3
2
2
+
11
2
2
Establish
the
deca
y
of
the
function
f
on
the
in
terv
al
3
2
;
+
∞
.
E.8519
Consider
the
function
f
defined
b
y
:
f
(
x
)
=
1
−
x
2
+
2
x
+
4
1
Establish
that
the
defining
set
of
f
is
:
D
f
=
R
\
1
−
5
;
1+
5
2
On
the
in
terv
al
1+
5
;
+
∞
,
establish
the
direction
of
the
function
f
.
6.
Unclassifie
d
financial
ye
ars
E.2296
W
e
define
the
function
f
on
R
whose
image
of
x
∈
R
is
defined
b
y
the
relation
:
f
(
x
)
=
8
x
2
−
2
x
+
1
1
Giv
e
the
canonical
form
of
the
function
f
.
2
Establish
that
the
function
f
is
minorized
b
y
7
8
.
3
a
Establish,
without
justification,
the
table
of
v
aria-
tions
of
the
function
f
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4409
sacados/4409
chapExoCorrec/4407
sacados/4407
chapExoCorrec/8516
sacados/8516
chapExoCorrec/8517
sacados/8517
chapExoCorrec/4481
sacados/4481
chapExoCorrec/8518
sacados/8518
chapExoCorrec/7252
sacados/7252
chapExoCorrec/7253
sacados/7253
chapExoCorrec/8519
sacados/8519
chapExoCorrec/2296
sacados/2296
-2
-1
2
3
I
-1
2
3
4
5
J
O
A
B
C
b
Deduce
that
the
function
f
admits
no
zero
at
R
.
E.205
Consider
the
function
f
whose
image
of
x
is
defined
b
y
the
relation
:
f
(
x
)
=
a
·
x
2
+
b
·
x
+
c
where
a
,
b
and
c
are
fixed
real
n
um
b
ers,
but
unkno
wn
at
presen
t.
Consider
the
represen
ta-
tion
of
the
function
f
in
the
orthogonal
frame
(
O
;
I
;
J
)
b
elo
w
:
1
Sho
w
that
the
n
um
b
ers
a
,
b
,
c
m
ust
v
erify
the
follo
wing
equation
system
:
a
−
b
+
c
=
9
2
a
+
b
+
c
=
1
2
4
a
+
2
b
+
c
=
3
2
Solv
e
the
previous
system
and
deriv
e
the
expression
of
f
(
x
)
.
E.4568
Consider
the
t
w
o
functions
f
whose
im-
age
of
a
n
um
b
er
x
is
giv
en
b
y
the
relation
:
f
(
x
)
=
−
3
·
x
2
+
3
·
x
+
6
1
Consider
the
p
olynomial
−
3
·
x
2
+3
·
x
+6
of
the
second
de-
gree
:
a
Using
the
calculator,
determine
the
t
w
o
ro
ots
of
this
p
olynomial.
b
Dra
w
up
the
sign
table
for
this
p
olynomial.
c
Deduce
the
definition
set
of
the
function
f
.
2
Using
the
calculator,
dra
w
up
the
table
of
v
ariations
of
the
function
f
.
E.5711
1
Establish
that
the
p
olynomial
2
x
2
−
3
x
−
2
admits
as
fac-
torized
form
:
2
x
2
−
3
x
−
2
=
2
·
x
+
a
b
·
x
+
c
where
the
v
alues
of
the
n
um
b
ers
a
,
b
,
c
are
to
b
e
sp
ecified.
2
Establish
that
the
p
olynomial
12
x
2
−
12
x
+3
admits
as
its
factorized
form
:
12
x
2
−
12
x
+
3
=
3
·
2
·
x
+
a
c
·
x
+
d
where
the
v
alues
of
the
n
um
b
ers
a
,
b
,
c
are
to
b
e
sp
ecified.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/205
sacados/205
-2
-1
2
3
I
-1
2
3
4
5
J
O
A
B
C
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sacados/4568
chapExoCorrec/5711
sacados/5711
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