what are integral roots please explain in detail do include how to solve such problems also find the integral roots - Maths - Polynomials

Question

what are integral roots?please explain
in detail
do include how to solve such
problems
also
find the integral roots of the polynomial
x3+6x2+11x+6

Ajanta Trivedi · Accepted Answer

integral roots are the roots which has integer values.

all the roots which are integer , is known as integral roots.

let the given polynomial be $p (x) = x^{3} + 6 x^{2} + 11 x + 6$

at x = -1

$p (- 1) = (- 1)^{3} + 6 * (- 1)^{2} + 11 * (- 1) + 6 = - 1 + 6 - 11 + 6 = 0$

thus x = -1 is a zero of p(x) and therefore (x+1) is a factor of p(x).

$x + 1 \begin{matrix} x^{2} + 5 x + 6 \\ x^{3} + 6 x^{2} + 11 x + 6 \end{matrix}$

$\underset{̲}{x^{3} + x^{2}} \times 5 x^{2} + 11 x \underset{̲}{5 x^{2} + 5 x} \times 6 x + 6 \underset{̲}{6 x + 6} \times$

now factorise $x^{2} + 5 x + 6$ .

$x^{2} + 5 x + 6 = x^{2} + 2 x + 3 x + 6 = x (x + 2) + 3 (x + 2) = (x + 2) (x + 3)$

therefore $p (x) = (x + 1) (x + 2) (x + 3)$

thus the zeroes of p(x) are -1, -2 and -3

and integral zeroes are -1, -2 and -3.

hope this helps you.

cheers!!

what are integral roots?please explain in detail do include how to solve such problems also find the integral roots of the polynomial x3+6x2+11x+6