use the rational zero theorem to find all the zeros of the polynomial function. Use the factor f over the real numbers.
f(x)=x^4+2x^3-7x^2-8x+12
x=
f(x)=
try factors of 12
on the first try, I got f(1) = 0
so x-1 is a factor
by long algebraic or by synthetic division , I got
x^4+2x^3-7x^2-8x+12
= (x-1)(x^3 + 3x^2 - 4x - 12)
grouping the x^3 + 3x^2 - 4x - 12
= x^2(x+3) - 4(x+3)
= (x+3)(x^2-4)
= (x+3)(x+2)(x-2)
so
x^4+2x^3-7x^2-8x+12
= (x-1)(x-2)(x+2)(x+3)
(had I been patient, I would have found
f(2) = 0, f(-3)=0 and f(-2) = 0 as well)
the zeros are
1, 2, -2, -3
To use the Rational Zero Theorem to find all the zeros of the polynomial function f(x) = x^4 + 2x^3 - 7x^2 - 8x + 12, we need to find the possible rational zeros.
The Rational Zero Theorem states that if a rational number p/q is a zero of the polynomial with integer coefficients, then p must be a factor of the constant term (in this case, 12), and q must be a factor of the leading coefficient (in this case, 1).
So, the possible rational zeros are determined by the factors of 12. They are: ±1, ±2, ±3, ±4, ±6, ±12.
Now, we can test each of these values by using synthetic division or long division to check if they are zeros of the polynomial. We are looking for values that give a remainder of 0 when the polynomial is divided by (x - value).
Using synthetic division, let's test if x = 1 is a zero:
1 | 1 2 -7 -8 12
-- 3 -4 -11 -19
-----------------
1 5 -11 -19 -7
Since the remainder is not 0, x = 1 is not a zero of the polynomial.
Continuing this process, we can test the other possible rational zeros (±2, ±3, ±4, ±6, ±12) until we find the zeros of the polynomial.
After testing all the possible rational zeros, we find that the polynomial function f(x) = x^4 + 2x^3 - 7x^2 - 8x + 12 has no rational zeros over the real numbers.