Question

p (x) = x3-2x2-5x+6

how many roots/Zeros will the polynomial have?
how do you know?

Answer 1

it is 3rd order ... highest exponent is 3 ... 3 roots

Answer 2

[AI-generated answer — Explain Bot]

To determine the number of roots or zeros a polynomial has, we need to use the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots in the complex number system (including real and imaginary roots).

In this case, we have a polynomial of degree 3: p(x) = x^3 - 2x^2 - 5x + 6. Therefore, we can conclude that this polynomial will have exactly 3 roots or zeros.

To verify this, we can also use the Rational Root Theorem. According to the theorem, if a polynomial with integer coefficients has a rational root p/q (where p and q are integers) in its simplest form, then p must be a factor of the constant term (in this case, 6) and q must be a factor of the leading coefficient (in this case, 1).

For the polynomial p(x) = x^3 - 2x^2 - 5x + 6, the constant term is 6 and the leading coefficient is 1. The factors of 6 are ±1, ±2, ±3, and ±6, and the factors of 1 are ±1.

By trying out these possible rational roots in the polynomial, we can determine if any of them are valid and find the zeros.