Question

x + x2 - 5x -5 = 0
state the possible rational zeros and find all the zeros of the polynomial

Answer 1

Answer:

The zeros of the polynomial are -1 and 5

Step-by-step explanation:

Quadratic Equation Solving

The standard representation of a quadratic equation is:

$ax^2+bx+c=0$

where a,b, and c are constants.

Solving with the quadratic formula:

$\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

We have the following equation to solve:

$x+x^2-5x-5=0$

Before attempting to solve it, we must simplify the equation.

Collecting like terms and reordering:

$x^2-4x-5=0$

Here: a=1, b=-4, c=-5

The discriminant of this quadratic equation is:

$d=b^2-4ac$

$d=(-4)^2-4(1)(-5)=16+20=36$

Given d is positive, the equation has two roots, and since d is a perfect square, both roots are rational.

Applying the formula:

$\displaystyle x=\frac{4\pm \sqrt{36}}{2(1)}$

$\displaystyle x=\frac{4\pm 6}{2}$

Dividing by 2:

$x=2\pm 3$

Separating both roots:

x = 2 + 3 = 5

x = 2 - 3 = -1

The zeros of the polynomial are -1 and 5