factorise ces polynômes.
a) $2m^2n - n$
$1^{\text{er}}$ facteur
$2^{\text{e}}$ facteur
résultat :
b) $-27a^2b - 6ab$
$1^{\text{er}}$ facteur
$2^{\text{e}}$ facteur
résultat :
c) $8x^3y^2 - 4xy^2z^2$
$1^{\text{er}}$ facteur
$2^{\text{e}}$ facteur
résultat :
d) $6a^4b^2c - 3a^3bc + 12a^2d^2 - 9abcd$
$1^{\text{er}}$ facteur
$2^{\text{e}}$ facteur
résultat :
e) $-10x^3y^3z^2 - 15x^3y^3z - 5x^3y^3$
$1^{\text{er}}$ facteur
$2^{\text{e}}$ facteur
f) $36s^4t^2v^2 + 54st^6v - 18s^4t^4v$
$1^{\text{er}}$ facteur
$2^{\text{e}}$ facteur

Question

Accepted Answer

### Part (a)
# Explanation:
## Step1: Find GCD of terms
For $2m^{2}n - n$, GCD of coefficients $2$ and $1$ is $1$, common variables: $n$ (since $n$ is in both terms, $m^{2}$ is only in first term). So $1^{\text{er}}$ facteur (first factor) is $n$.
## Step2: Divide each term by GCD
Divide $2m^{2}n$ by $n$ gives $2m^{2}$, divide $-n$ by $n$ gives $-1$. So $2^{\text{e}}$ facteur (second factor) is $2m^{2}-1$.
## Step3: Write factored form
Result: $n(2m^{2}-1)$

# Answer:
$1^{\text{er}}$ facteur: $n$
$2^{\text{e}}$ facteur: $2m^{2}-1$
Résultat: $n(2m^{2}-1)$

### Part (b)
# Explanation:
## Step1: Find GCD of terms
For $-27a^{2}b - 6ab$, GCD of coefficients $27$ and $6$ is $3$ (take absolute values), common variables: $ab$ (both terms have $a$ and $b$, $a^{2}$ is only in first term). Also, note the negative sign, we can factor out $-3ab$ (or $3ab$, but let's check). Coefficients: $-27$ and $-6$, GCD of $27$ and $6$ is $3$, so factor out $-3ab$ (to make first term positive inside). Wait, or $3ab$ with negative. Let's see: $-27a^{2}b=-3ab\times9a$, $-6ab=-3ab\times2$. So $1^{\text{er}}$ facteur is $-3ab$ (or $3ab$, but let's do it properly). GCD of $-27$ and $-6$ is $-3$? No, GCD is positive, but we can factor out $-3ab$. Wait, coefficients: $27$ and $6$, GCD is $3$, variables: $ab$ (since $a^{2}b$ and $ab$, common is $ab$). So $1^{\text{er}}$ facteur: $3ab$ (wait, $-27a^{2}b = 3ab\times(-9a)$, $-6ab=3ab\times(-2)$). So $1^{\text{er}}$ facteur: $3ab$ (or $-3ab$, but let's check signs). Let's factor out $-3ab$: $-27a^{2}b - 6ab=-3ab(9a + 2)$. Wait, no: $-27a^{2}b\div(-3ab)=9a$, $-6ab\div(-3ab)=2$. So $1^{\text{er}}$ facteur: $-3ab$, $2^{\text{e}}$ facteur: $9a + 2$. Wait, maybe better to take positive GCD. GCD of $27$ and $6$ is $3$, common variables $ab$, so factor out $3ab$ with negative: $-3ab(9a + 2)$. Wait, let's recalculate: $-27a^{2}b - 6ab = -3ab(9a + 2)$ (because $-3ab\times9a=-27a^{2}b$, $-3ab\times2=-6ab$). So $1^{\text{er}}$ facteur: $-3ab$ (or $3ab$ with negative inside, but let's do it as GCD of coefficients $27$ and $6$ is $3$, variables $ab$, so factor out $3ab$ and handle signs. Alternatively, factor out $3ab$ from $-27a^{2}b - 6ab$: $3ab(-9a - 2)$, but maybe factor out $-3ab$ to make the second factor have positive first term: $-3ab(9a + 2)$. So $1^{\text{er}}$ facteur: $-3ab$, $2^{\text{e}}$ facteur: $9a + 2$.
## Step2: Verify
Multiply $-3ab(9a + 2)=-27a^{2}b - 6ab$, correct.
## Step3: Write factored form
Result: $-3ab(9a + 2)$ (or $3ab(-9a - 2)$, but first form is better with positive inside the parentheses if possible, but let's check the GCD again. Coefficients: $-27$ and $-6$, GCD is $3$ (since GCD is positive), common variables $ab$, so factor out $3ab$ with a negative sign: $3ab(-9a - 2)$, but maybe the problem expects factoring out the greatest common divisor with sign to make the second factor have positive leading coefficient. So $1^{\text{er}}$ facteur: $3ab$ (no, wait, $-27a^{2}b = 3ab\times(-9a)$, $-6ab=3ab\times(-2)$, so $1^{\text{er}}$ facteur is $3ab$, $2^{\text{e}}$ facteur is $-9a - 2$, but that's also correct. Alternatively, factor out $-3ab$, then $2^{\text{e}}$ facteur is $9a + 2$. Let's go with $-3ab$ as first factor.

# Answer:
$1^{\text{er}}$ facteur: $-3ab$ (or $3ab$ with adjustment, but let's confirm: $-27a^{2}b - 6ab=-3ab(9a + 2)$)
$2^{\text{e}}$ facteur: $9a + 2$
Résultat: $-3ab(9a + 2)$

### Part (c)
# Explanation:
## Step1: Find GCD of terms
For $8x^{3}y^{2}-4xy^{2}z^{2}$, GCD of coefficients $8$ and $4$ is $4$, common variables: $xy^{2}$ (both terms have $x$, $y^{2}$; $x^{3}$ is only in first term, $z^{2}$ only in second term). So $1^{\text{er}}$ facteur is $4xy^{2}$.
## Step2: Divide each term by GCD
Divide $8x^{3}y^{2}$ by $4xy^{2}$ gives $2x^{2}$, divide $-4xy^{2}z^{2}$ by $4xy^{2}$ gives $-z^{2}$. So $2^{\text{e}}$ facteur is $2x^{2}-z^{2}$.
## Step3: Write factored form
Result: $4xy^{2}(2x^{2}-z^{2})$

# Answer:
$1^{\text{er}}$ facteur: $4xy^{2}$
$2^{\text{e}}$ facteur: $2x^{2}-z^{2}$
Résultat: $4xy^{2}(2x^{2}-z^{2})$

### Part (d)
# Explanation:
## Step1: Find GCD of terms
For $6a^{4}b^{2}c - 3a^{3}bc + 12a^{2}d^{2}-9abcd$, GCD of coefficients $6,3,12,9$ is $3$, common variables: $a$ (all terms have $a$, $b$ is in first, second, fourth; $c$ is in first, second, fourth; $d$ is in third, fourth). Wait, let's check each term:
- $6a^{4}b^{2}c = 3a\times2a^{3}b^{2}c$
- $-3a^{3}bc = 3a\times(-a^{2}bc)$
- $12a^{2}d^{2}=3a\times4ad^{2}$
- $-9abcd = 3a\times(-3bcd)$

So common factor is $3a$. Thus, $1^{\text{er}}$ facteur is $3a$.
## Step2: Divide each term by GCD
Divide $6a^{4}b^{2}c$ by $3a$ gives $2a^{3}b^{2}c$, divide $-3a^{3}bc$ by $3a$ gives $-a^{2}bc$, divide $12a^{2}d^{2}$ by $3a$ gives $4ad^{2}$, divide $-9abcd$ by $3a$ gives $-3bcd$. So $2^{\text{e}}$ facteur is $2a^{3}b^{2}c - a^{2}bc + 4ad^{2}-3bcd$.
## Step3: Write factored form
Result: $3a(2a^{3}b^{2}c - a^{2}bc + 4ad^{2}-3bcd)$

# Answer:
$1^{\text{er}}$ facteur: $3a$
$2^{\text{e}}$ facteur: $2a^{3}b^{2}c - a^{2}bc + 4ad^{2}-3bcd$
Résultat: $3a(2a^{3}b^{2}c - a^{2}bc + 4ad^{2}-3bcd)$

### Part (e)
# Explanation:
## Step1: Find GCD of terms
For $-10x^{3}y^{3}z^{2}-15x^{3}y^{3}z - 5x^{3}y^{3}$, GCD of coefficients $10,15,5$ is $5$, common variables: $x^{3}y^{3}$ (all terms have $x^{3}y^{3}$; $z^{2}$ in first, $z$ in second, none in third). Also, note the negative signs, factor out $-5x^{3}y^{3}$ (to make first term positive inside? Wait, coefficients: $-10,-15,-5$, GCD is $5$ (positive), but let's factor out $-5x^{3}y^{3}$:

- $-10x^{3}y^{3}z^{2}=-5x^{3}y^{3}\times2z^{2}$
- $-15x^{3}y^{3}z=-5x^{3}y^{3}\times3z$
- $-5x^{3}y^{3}=-5x^{3}y^{3}\times1$

So $1^{\text{er}}$ facteur is $-5x^{3}y^{3}$ (or $5x^{3}y^{3}$ with negative inside, but let's do it as $-5x^{3}y^{3}$).
## Step2: Divide each term by GCD
Divide $-10x^{3}y^{3}z^{2}$ by $-5x^{3}y^{3}$ gives $2z^{2}$, divide $-15x^{3}y^{3}z$ by $-5x^{3}y^{3}$ gives $3z$, divide $-5x^{3}y^{3}$ by $-5x^{3}y^{3}$ gives $1$. So $2^{\text{e}}$ facteur is $2z^{2}+3z + 1$.
## Step3: Write factored form
Result: $-5x^{3}y^{3}(2z^{2}+3z + 1)$ (we can also factor $2z^{2}+3z + 1=(2z + 1)(z + 1)$, but the problem just asks for factoring out the GCD, so this is sufficient).

# Answer:
$1^{\text{er}}$ facteur: $-5x^{3}y^{3}$
$2^{\text{e}}$ facteur: $2z^{2}+3z + 1$
Résultat: $-5x^{3}y^{3}(2z^{2}+3z + 1)$

### Part (f)
# Explanation:
## Step1: Find GCD of terms
For $36s^{4}t^{2}v^{2}+54st^{6}v - 18s^{4}t^{4}v$, GCD of coefficients $36,54,18$ is $18$, common variables: $s$ (first and third have $s^{4}$, second has $s$; $t$ (all have $t$; $v$ (first, second, third have $v$). Let's check each term:
- $36s^{4}t^{2}v^{2}=18st v\times2s^{3}t v$
- $54st^{6}v=18st v\times3t^{5}$
- $-18s^{4}t^{4}v=18st v\times(-s^{3}t^{3})$

Wait, no: GCD of $s$ terms: minimum power of $s$ is $1$ (in second term), minimum power of $t$ is $1$ (in first term), minimum power of $v$ is $1$ (all terms). Coefficients: GCD of $36,54,18$ is $18$. So common factor is $18stv$. Let's verify:
- $36s^{4}t^{2}v^{2}\div18stv = 2s^{3}tv$
- $54st^{6}v\div18stv = 3t^{5}$
- $-18s^{4}t^{4}v\div18stv = -s^{3}t^{3}$

So $1^{\text{er}}$ facteur is $18stv$.
## Step2: Divide each term by GCD
As above, the second factor is $2s^{3}tv + 3t^{5}-s^{3}t^{3}$ (wait, no: $36s^{4}t^{2}v^{2}\div18stv = 2s^{3}t v$, $54st^{6}v\div18stv = 3t^{5}$, $-18s^{4}t^{4}v\div18stv = -s^{3}t^{3}$. So $2^{\text{e}}$ facteur is $2s^{3}tv + 3t^{5}-s^{3}t^{3}$.
## Step3: Write factored form
Result: $18stv(2s^{3}tv + 3t^{5}-s^{3}t^{3})$ (we can factor further inside, but the problem asks for factoring out the GCD, so this is okay).

# Answer:
$1^{\text{er}}$ facteur: $18stv$
$2^{\text{e}}$ facteur: $2s^{3}tv + 3t^{5}-s^{3}t^{3}$
Résultat: $18stv(2s^{3}tv + 3t^{5}-s^{3}t^{3})$

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step1: Find GCD of terms

Step2: Divide each term by GCD

Step3: Write factored form

Step1: Find GCD of terms

Step2: Verify

Step3: Write factored form

Step1: Find GCD of terms

Step2: Divide each term by GCD

Step3: Write factored form

Answer:

Part (b)