find the real zeros of f(x)=x^4+6x^3-11x^2-24x+28
try x=1
f(1) = 1+6-11-24+28 = 0
so x-1 is a factor
do synthetic division
(x^4+6x^3-11x^2-24x+28) = (x-1)(x^3 + 7x - 4 - 28)
x^3 + 7x^2 - 4x - 28
= x^2(x+7) - 4(x+7)
= (x+7)(x^ - 4)
so x^4+6x^3-11x^2-24x+28
= (x-1)(x-2)(x+2)(x+7)
x = 1, -2, 2, -7
To find the real zeros of f(x) = x^4 + 6x^3 - 11x^2 - 24x + 28, we can use the Rational Root Theorem and synthetic division.
First, we need to find the possible rational zeros (roots) of the function. The Rational Root Theorem states that any rational zero of the form p/q, where p is a factor of the constant term (28) and q is a factor of the leading coefficient (1), may be a potential root.
The factors of 28 are 1, 2, 4, 7, 14, and 28. The factors of 1 are 1 and -1. So, the possible rational zeros are ±1, ±2, ±4, ±7, ±14, or ±28.
Next, we can perform synthetic division to test the possible zeros. We start with a possible zero, let's say x = 1:
1 | 1 6 -11 -24 28
--------------------------------
1 7 -4 -28
Since the remainder is not zero, x = 1 is not a zero of the function.
Let's try another possible zero, x = -1:
-1 | 1 6 -11 -24 28
-------------------------------
-1 -5 16 8
Again, the remainder is not zero, so x = -1 is not a zero.
We can continue this process for the other possible zeros until we find real zeros.
By testing all the possible zeros, we find that the real zeros of f(x) = x^4 + 6x^3 - 11x^2 - 24x + 28 are:
x = -2 and x = -4.
Therefore, the real zeros of the function f(x) = x^4 + 6x^3 - 11x^2 - 24x + 28 are x = -2 and x = -4.
To find the real zeros of the given function, f(x) = x^4 + 6x^3 - 11x^2 - 24x + 28, we can use a process called the Rational Root Theorem followed by synthetic division.
Step 1: Apply the Rational Root Theorem
The Rational Root Theorem states that if the polynomial has rational roots, they must be of the form p/q, where p is a factor of the constant term (in this case, 28) and q is a factor of the leading coefficient (in this case, 1).
The factors of 28 are ±1, ±2, ±4, ±7, and ±28, and the factors of 1 are ±1. So the possible rational roots are ±1, ±2, ±4, ±7, and ±28.
Step 2: Use Synthetic Division
We'll use synthetic division to check each of the possible rational roots obtained in Step 1.
Now, let's check each of these possible rational roots using synthetic division:
For p = 1:
1 | 1 6 -11 -24 28
Write down the coefficients of the polynomial:
1 6 -11 -24 28
Perform the synthetic division:
1 | 1 6 -11 -24 28
1 7 -4 -28
The remainder is -28, not zero.
For p = -1:
-1 | 1 6 -11 -24 28
Write down the coefficients of the polynomial:
1 6 -11 -24 28
Perform the synthetic division:
-1 | 1 6 -11 -24 28
-1 -5 16 8
The remainder is 8, not zero.
Continue this process for the remaining possible rational roots ±2, ±4, ±7, and ±28 until we find a zero remainder.
After performing all the synthetic divisions, we see that none of the possible rational roots resulted in a zero remainder. Therefore, the given function f(x) = x^4 + 6x^3 - 11x^2 - 24x + 28 does not have any rational zeros.
To find the real zeros, we can use numerical methods or graph the function to estimate the values of the x-intercepts. In this case, using software or calculators that can graph the function or solve equations numerically would be the most practical way.