Question

exercice

1. effectue les divisions suivantes et donne le quotient. dans chaque cas, le diviseur est non nul.

a) $(9t^2 - t - 12) \div (3t + 4)$
b) $(20y^4 - 15y^3 - 5y^2) \div (5y - 5)$
c) $(m^4 + 2m^2 - 8) \div (m^2 - 2)$
d) $(18d^2 - 3d - 6) \div (3d - 2)$
e) $(10x^3 - x^2 + 6x - 4) \div (2x - 1)$
f) $(e^2 - 2e - 1) \div (e + 1)$

Explanation:

Step1: Divide polynomials (a)

Divide $6t^2 - t - 12$ by $3t + 4$.
Use polynomial long division:
$$\frac{6t^2 - t - 12}{3t + 4} = 2t - 3$$
Check: $(3t+4)(2t-3)=6t^2-9t+8t-12=6t^2-t-12$

Step2: Divide polynomials (b)

Divide $20y^4 - 15y^3 - 5y^2$ by $5y - 5$.
Factor numerator: $5y^2(4y^2 - 3y - 1)=5y^2(4y+1)(y-1)$
Factor denominator: $5(y-1)$
Cancel common terms:
$$\frac{5y^2(4y+1)(y-1)}{5(y-1)} = y^2(4y+1)=4y^3 + y^2$$

Step3: Divide polynomials (c)

Divide $m^4 + 2m^2 - 8$ by $m^2 - 2$.
Use polynomial long division:
$$\frac{m^4 + 2m^2 - 8}{m^2 - 2} = m^2 + 4$$
Check: $(m^2-2)(m^2+4)=m^4+4m^2-2m^2-8=m^4+2m^2-8$

Step4: Divide polynomials (d)

Divide $18d^2 - 3d - 6$ by $3d - 2$.
Use polynomial long division:
$$\frac{18d^2 - 3d - 6}{3d - 2} = 6d + 3$$
Check: $(3d-2)(6d+3)=18d^2+9d-12d-6=18d^2-3d-6$

Step5: Divide polynomials (e)

Divide $10x^3 - x^2 + 6x - 4$ by $2x - 1$.
Use polynomial long division:
$$\frac{10x^3 - x^2 + 6x - 4}{2x - 1} = 5x^2 + 2x + 4$$
Check: $(2x-1)(5x^2+2x+4)=10x^3+4x^2+8x-5x^2-2x-4=10x^3-x^2+6x-4$

Step6: Divide polynomials (f)

Divide $e^2 - 2e - 1$ by $e + 1$.
Use polynomial long division:
$$\frac{e^2 - 2e - 1}{e + 1} = e - 3 + \frac{2}{e+1}$$
Quotient is the polynomial part: $e - 3$

Answer:

a) $\boldsymbol{2t - 3}$
b) $\boldsymbol{4y^3 + y^2}$
c) $\boldsymbol{m^2 + 4}$
d) $\boldsymbol{6d + 3}$
e) $\boldsymbol{5x^2 + 2x + 4}$
f) $\boldsymbol{e - 3}$