Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers f(x)=9x^4+28x^3+66x^2+196x+21
Find the real zeros of f
Use the real zeros to factor f
f(x)=
To find the real zeros of the polynomial function f(x) = 9x^4 + 28x^3 + 66x^2 + 196x + 21 using the rational zeros theorem, we need to consider the factors of the constant term (21) divided by the factors of the leading coefficient (9).
The factors of 21 are ±1, ±3, ±7, ±21, and the factors of 9 are ±1, ±3, ±9.
Possible rational zeros are obtained by dividing all the possible combinations of the factors of 21 by the factors of 9.
Possible rational zeros: ±1/1, ±1/3, ±1/9, ±3/1, ±3/3, ±3/9, ±7/1, ±7/3, ±7/9, ±21/1, ±21/3, ±21/9
We can now use these potential rational zeros to check whether they are actually zeros of the polynomial by plugging them into f(x) and seeing whether the result is zero.
By substituting these values into f(x), we find that the real zeros of f(x) are x = -1 and x = -7/3.
Next, we will use synthetic division or long division to divide f(x) by these zeros to factor f over the real numbers.
Using synthetic division with x = -1:
1 |-1 28 66 196 21
| -1 -27 -39 -157
-------------------
-1 27 39 157 -136
The quotient is -x^3 + 27x^2 + 39x + 157, and the remainder is -136.
Using synthetic division with x = -7/3:
-7/3 | -1 27 39 157 -136
| 7 -20 -57 0
-------------------
-1 34 19 100 -136
The quotient is -x^2 + 34x + 19, and the remainder is -136.
Therefore, factoring f(x) over the real numbers, we have:
f(x) = (x + 1)(-x^3 + 27x^2 + 39x + 157) = (x + 1)(-x^2 + 34x + 19)
To find the real zeros of the polynomial function f(x) = 9x^4 + 28x^3 + 66x^2 + 196x + 21, we can use the rational zeros theorem.
The rational zeros theorem states that if a polynomial has a rational zero p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), then p/q will be a zero of the polynomial.
Step 1: Identify the factors of the constant term (21) and the leading coefficient (9).
Factors of 21: ±1, ±3, ±7, ±21
Factors of 9: ±1, ±3, ±9
Step 2: Write down all possible rational zeros, which are the combinations of the factors identified in Step 1.
Possible rational zeros: ±1, ±3, ±7, ±21, ±1/3, ±7/3, ±1/9, ±7/9
Step 3: Test each possible zero by substituting it into the polynomial function f(x) and checking if it equals zero.
Upon testing, we find that -1 is a zero of the function f(x) = 9x^4 + 28x^3 + 66x^2 + 196x + 21.
Therefore, one of the real zeros of f is x = -1.
Step 4: Divide the polynomial f(x) by (x + 1) using synthetic division to obtain the remaining factor.
Performing the synthetic division:
-1 | 9 28 66 196 21
| -9 -19 -47 -149
|———————————————
9 19 47 47 -128
The resulting quotient is 9x^3 + 19x^2 + 47x + 47, and the remainder is -128.
So, f(x) can be factored as (x + 1)(9x^3 + 19x^2 + 47x + 47).
Therefore, the real zeros of f(x) are x = -1.
The factored form of f(x) over the real numbers is f(x) = (x + 1)(9x^3 + 19x^2 + 47x + 47).