Exercices sur le second degré 1S Factorisation du trinôme.

Exercice 1

Factoriser les polynômessuivants :

I1. Factoriser R(t) = 162t²−50^{à l'aided'une identité} remarquable.

I2. R(t) =t²+ 10t+ 16 I3. S(x) =−33x²+ 10x+ 8 I4. Q(y) =y²+ 1

Exercice 2

Factoriser les polynômessuivants :

I1. Factoriser Q(t) = 320t²+ 320t+ 80^{à l'aided'uneidentité} remarquable.

I2. R(x) =x²+ 7x+ 10 I3. P(z) =−9z²−21z−10 I4. R(y) =−y²+ 3y+ 4 Exercice 3

Factoriser les polynômessuivants :

I1. Factoriser S(x) = 81x²−18x+ 1 ^{à l'aided'uneidentité} remarquable.

I2. Q(x) =x²+x−20 I3. R(z) =−2z²−13z−21 I4. R(z) =−z²+ 8z+ 6 Exercice 4

Factoriser les polynômessuivants :

I1. Factoriser R(x) = 64x²+ 16x+ 1^{àl'aide d'uneidentité} remarquable.

I2. R(t) =t²+ 3t−4 I3. S(t) =−4t²+ 7t−3 I4. Q(t) =−t²+ 9t+ 6

Corrigé de l’exercice 1

I1. Factoriser R(t) = 162t²−50

162t²−50 = 2×

81x²−25

= 2×

(9t)²−5²

= 2(9t+ 5)(9t−5) I2. Factoriser R(t) =t²+ 10t+ 16

Je calcule∆ = 10²−4×1×16 = 36 ^et√

36 = 6^.

Comme∆>0^,R(t) ^{adeuxracines :}

−10−√ 36

2×1 =−10−√ 36 2

−10 +√ 36

2×1 =−10 +√ 36 2

=−10−6

2 =−10 + 6

2

=−16

2 =−4

2

=−8 =−2

Lesracinesde R ^sont t₁=−8^ett₂ =−2^.

Onpeutdonc écrire

R(t) = (t−(−8)) (t−(−2)) = (t+ 8) (t+ 2) I3. Factoriser S(x) =−33x²+ 10x+ 8

Je calcule∆ = 10²−4×(−33)×8 = 1 156 ^et√

1 156 = 34^.

Comme∆>0^,S(x)^{a deuxracines :}

−10 +√ 1 156

2×(−33) =−10 +√ 1 156

−66

−10−√ 1 156

2×(−33) =−10−√ 1 156

−66

=−10 + 34

−66 =−10−34

−66

= 24

−66 =−44

−66

=−4×(−6)

11×(−6)

=2×(−22)

3×(−22)

=−4

11 =2

3

Lesracinesde S ^sont x₁= −4

11 ^etx₂= 2 3^.

Onpeutdonc écrire

S(x) =−33×

x−

−4

11 x−2 3

=−33×

x+ 4

11 x−2 3

I4. Factoriser Q(y) =y²+ 1

Je calcule∆ = 0²−4×1×1 =−4^.

Comme∆<0^,Q(y) ^{n'apasde racines.Onne peutpasfactoriser}Q(y)^. Corrigé de l’exercice 2

I1. Factoriser Q(t) = 320t²+ 320t+ 80 320t²+ 320t+ 80 = 80

4x²+ 4x+ 1

= 80

(2t)²+ 2 2t 1 + 1²

= 80(2t+ 1)²

R(x) =x + 7x+ 10

Je calcule∆ = 7²−4×1×10 = 9 ^et√ 9 = 3^.

Comme∆>0^,R(x) ^{a deuxracines:}

−7−√ 9

2×1 =−7−√ 9 2

−7 +√ 9

2×1 =−7 +√ 9 2

=−7−3

2 =−7 + 3

2

=−10

2 =−4

2

=−5 =−2

Lesracinesde R ^sont x₁ =−5^et x₂=−2^.

Onpeutdonc écrire

R(x) = (x−(−5)) (x−(−2)) = (x+ 5) (x+ 2) I3. Factoriser P(z) =−9z²−21z−10

Je calcule∆ = (−21)²−4×(−9)×(−10) = 81 ^et√

81 = 9^.

Comme∆>0^,P(x) ^{a deuxracines:}

−(−21) +√ 81

2×(−9) =21 +√ 81

−18

−(−21)−√ 81

2×(−9) =21−√ 81

−18

=21 + 9

−18 =21−9

−18

= 30

−18 = 12

−18

=−5×(−6)

3×(−6)

=−2×(−6)

3×(−6)

=−5

3 =−2

3

Lesracinesde P ^sont x₁ = −5

3 ^etx₂= −2 3 ^.

Onpeutdonc écrire

P(x) =−9×

x−

−5

3 x−

−2 3

=−9×

x+5

3 x+ 2 3

I4. Factoriser R(y) =−y²+ 3y+ 4

Je calcule∆ = 3²−4×(−1)×4 = 25^et√

25 = 5^.

Comme∆>0^,R(y)^{a deuxracines:}

−3 +√ 25

2×(−1) =−3 +√ 25

−2

−3−√ 25

2×(−1) =−3−√ 25

−2

=−3 + 5

−2 =−3−5

−2

= 2

−2 =−8

−2

=−1 =4

Lesracinesde R ^sont y₁=−1 ^ety₂ = 4^.

Onpeutdonc écrire

R(y) =−1×(y−(−1)) (y−4) =−1×(y+ 1) (y−4)

Corrigé de l’exercice 3

I1. Factoriser S(x) = 81x²−18x+ 1

81x²−18x+ 1 = (9x)²−2×9x×1 + 1² = (9x−1)² I2. Factoriser Q(x) =x²+x−20

Je calcule∆ = 1²−4×1×(−20) = 81^et √

81 = 9^.

Comme∆>0^,Q(x)^{a deuxracines :}

−1−√ 81

2×1 =−1−√ 81 2

−1 +√ 81

2×1 =−1 +√ 81 2

=−1−9

2 =−1 + 9

2

=−10

2 =8

2

=−5 =4

Lesracinesde Q^sont x₁=−5 ^etx₂ = 4^.

Onpeutdonc écrire

Q(x) = (x−(−5)) (x−4) = (x+ 5) (x−4) I3. Factoriser R(z) =−2z²−13z−21

Je calcule∆ = (−13)²−4×(−2)×(−21) = 1^.

Comme∆>0^,R(x) ^{a deuxracines:}

−(−13) +√ 1

2×(−2) =13 +√ 1

−4

−(−13)−√ 1

2×(−2) =13−√ 1

−4

=13 + 1

−4 =13−1

−4

=14

−4 =12

−4

=−7×(−2)

2×(−2)

=−3

=−7 2

Lesracinesde R ^sont x₁ = −7

2 ^etx₂ =−3^.

Onpeutdonc écrire

R(x) =−2×

x−

−7 2

(x−(−3)) =−2×

x+7 2

(x+ 3) I4. Factoriser R(z) =−z²+ 8z+ 6

Je calcule∆ = 8²−4×(−1)×6 = 88^et√

88 = 2√ 22^.

Comme∆>0^,R(z) ^{a deuxracines:}

−8 +√ 88

2×(−1) =−8 +√ 88

−2

−8−√ 88

2×(−1) =−8−√ 88

−2

=−8 + 2√ 22

−2 =−8−2√

22

−2

=4×(−2)−1×(−2)

√22 1×(−2)

=4×(−2)+ 1×(−2)

√22 1×(−2)

=4 √

22 =4 +√

22

R z₁ = 4− 22 z₂= 4 + 22

Onpeutdonc écrire

R(z) =−1×

z−

4−√

22 z−

4 +√ 22 Corrigé de l’exercice 4

I1. Factoriser R(x) = 64x²+ 16x+ 1

64x²+ 16x+ 1 = (8x)²+ 2×8x×1 + 1² = (8x+ 1)² I2. Factoriser R(t) =t²+ 3t−4

Je calcule∆ = 3²−4×1×(−4) = 25^et√

25 = 5^.

Comme∆>0^,R(t) ^{adeuxracines :}

−3−√ 25

2×1 =−3−√ 25 2

−3 +√ 25

2×1 =−3 +√ 25 2

=−3−5

2 =−3 + 5

2

=−8

2 =2

2

=−4 =1

Lesracinesde R ^sont t₁=−4^ett₂ = 1^.

Onpeutdonc écrire

R(t) = (t−(−4)) (t−1) = (t+ 4) (t−1) I3. Factoriser S(t) =−4t²+ 7t−3

Je calcule∆ = 7²−4×(−4)×(−3) = 1^.

Comme∆>0^,S(x)^{a deuxracines :}

−7 +√ 1

2×(−4) =−7 +√ 1

−8

−7−√ 1

2×(−4) =−7−√ 1

−8

=−7 + 1

−8 =−7−1

−8

=−6

−8 =−8

−8

=3×(−2)

4×(−2)

=1

=3 4

Lesracinesde S ^sont x₁= 3

4 ^etx₂ = 1^.

Onpeutdonc écrire

S(x) =−4×

x−3 4

(x−1) I4. Factoriser Q(t) =−t²+ 9t+ 6

Je calcule∆ = 9²−4×(−1)×6 = 105^.

Comme∆>0^,Q(t) ^{a deuxracines:}

−9 +√ 105

2×(−1) =−9 +√ 105

−2

−9−√ 105

2×(−1) =−9−√ 105

−2

=9×(−1)−1×(−1)

√105 2×(−1)

=9×(−1)+ 1×(−1)

√105 2×(−1)

=9−√ 105

2 =9 +√

105 2

Lesracinesde Q^sont t₁ = 9−√ 105

2 ^ett₂ = 9 +√ 105

2 ^.

Onpeutdonc écrire

Q(t) =−1× t−9−√ 105 2

!

t− 9 +√ 105 2

!