what are all the real zeros for 1x^5+2x^4+4x^3+19x^2+9x-5?
To find the zeros of a polynomial, we can use a technique called the Rational Root Theorem.
The Rational Root Theorem states that if a polynomial equation has a rational root (a root that can be expressed as a fraction), then it will be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
In your polynomial equation 1x^5+2x^4+4x^3+19x^2+9x-5, the leading coefficient is 1, and the constant term is -5.
So, the possible rational roots for this equation are the factors of -5 (constant term) divided by the factors of 1 (leading coefficient).
The factors of -5 are ±1 and ±5.
The factors of 1 are ±1.
Therefore, the possible rational roots are: ±1/1, ±5/1, which simplify to ±1 and ±5.
Now, we can use these potential roots and apply synthetic division or polynomial long division to find the real zeros.
By trying each potential root, we can determine which ones actually give us zeros.
Using synthetic division or long division, we find that the real zeros of the polynomial equation 1x^5+2x^4+4x^3+19x^2+9x-5 are:
x = -5, -1, 1
So, the real zeros for the given polynomial are -5, -1, and 1.