List all possible (or potential) rational zeros for the polynomial below. Find all real zeros of the polynomial below and factor completely over the real numbers. Please show all of your work.
f(x) = x^4 - 7x^3 - 3x^2 + 19x + 14
Potential rational zeros are:
+-1,+-2,+-7,+-14
f(-1)=1+7-3-19+14=0
f(x)=(x+1)g(x), g(x)=x^3-8x^2+5x+14
g(-1)=-1-8-5+14=0
g(x)=(x+1)h(x),
f(x)=(x+1)^2*h(x), h(x)=x^2-9x+14
h(2)=4-18+14=0
h(x)=(x-2)(x-7)
f(x)=(x-2)(x-7)(x+1)^2
UNDEFINED!
2x^5+9x^3+3x^2-6
To find the possible rational zeros of a polynomial, we can use the rational root theorem. The rational root theorem states that if a polynomial has a rational root, it must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
For the polynomial f(x) = x^4 - 7x^3 - 3x^2 + 19x + 14, the constant term is 14, and the leading coefficient is 1. Therefore, the possible rational zeros can be found by listing all the factors of 14 and dividing them by the factors of 1.
The factors of 14 are ±1, ±2, ±7, ±14, and the factors of 1 are ±1. So, the possible rational zeros are given by the following fractions:
±1/1, ±2/1, ±7/1, ±14/1
Simplifying these fractions, we get the following possible rational zeros:
±1, ±2, ±7, ±14
Now, to find the real zeros of the polynomial, we can use either synthetic division or long division to factorize the polynomial.
Let's use long division:
_________
x^4 - 7x^3 - 3x^2 + 19x + 14 | x^4 + 0x^3 - 3x^2 + 19x + 14
- (x^4 - 7x^3 - 3x^2 + 19x + 14)
________________
7x^3 + 16x^2
- (7x^3 - 49x^2 - 42x)
____________________
65x^2 + 61x
- (65x^2 - 455x - 390)
____________________
516x + 404
- (516x + 360)
______________
44
The remainder is 44, which means that the polynomial does not evenly divide by x - k, where k is any of the possible rational zeros. So, none of the possible rational zeros are real zeros of the polynomial.
Therefore, the polynomial x^4 - 7x^3 - 3x^2 + 19x + 14 does not have any real zeros.
To factor the polynomial completely over the real numbers, we can use a different approach. Since there are no real roots, we cannot factor the polynomial as a product of linear factors. So, the polynomial is irreducible over the real numbers and cannot be factored any further.