Question

. factorise: x^3 + 3x^2 - 13x -15

Answer 1

Step-by-step explanation:

$\sf x^3 + 3x^2 - 13x - 15$

By Rational root theorem, all rational roots of a polynomial are in the form of $\dfrac{p}{q}$ , where $p$ divides the constant term $-15$ and $q$ divides the leading coefficient $1$. One such root is $-5$ Factor of the polynomial dividing it by $x+5$.

$\sf (x+5)(x^2 -2x -3)$

Consider $(x^2-2x-3)$. Factor the expression by grouping. First, the expression needs to be rewritten as $x^2 + ax + bx - 3$. To find $a$ and $b$, set up a system to be solved.

$a+b=-2$

$ab=1(-3)=-3$

Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$a=-3$

$b=1$

Rewrite $x^2-2x-3$ as $(x^2-3x)+(x-3)$

$\sf (x^2 - 3x)+(x-3)$

Factor out $x$ in $x^2-3x$.

$\sf x(x-3)+(x-3)$

Factor out common term $x-3$ using distributive property.

$\sf (x-3)(x+1)$

We have completely factorised $x^2-2x-3$

$\sf (x+5)(x-3)(x+1)$