Factorization, Ruffini’s rule and polynomial division

Expand

$$(2x-5{)}^2\text{and}(x+3{)}^3.$$

Solution available to those who have unlocked the subject.

Factor

$$9x^2-16\text{and}{x}^4-81.$$

Solution available to those who have unlocked the subject.

Factor

$$x^3+8\text{and}27x^3-1.$$

Solution available to those who have unlocked the subject.

Factor

$$12x^3{y}^2-18x^2{y}^3+6x^2{y}^2.$$

Solution available to those who have unlocked the subject.

Find the quotient and the remainder of the division of

$$x^3-4x^2+x+6\text{by}x-2.$$

Solution available to those who have unlocked the subject.

Find the quotient and the remainder of the division of

$$2x^4+x^3-3x^2+x-1\text{by}{x}^2+1.$$

Solution available to those who have unlocked the subject.

Compute, without carrying out the divisions, the remainder of the division of

$$P(x)=x^5-2x^3+x-4$$

by $x-1$ and by $x+2$.

Solution available to those who have unlocked the subject.

Find $k$ so that $x-3$ divides

$$P(x)=x^3-kx+6.$$

Solution available to those who have unlocked the subject.

Factor completely

$$x^3+x^2-10x+8.$$

Solution available to those who have unlocked the subject.

List the rational candidates and factor completely

$$3x^3-2x^2-3x+2.$$

Solution available to those who have unlocked the subject.

Factor

$$x^3+4x^2+3x+12.$$

Solution available to those who have unlocked the subject.

Factor completely

$$x^4-13x^2+36.$$

Solution available to those who have unlocked the subject.