QuestionFind the real zeros of the polynomial using the rational zeros theorem.
Studdy Solution
STEP 1
Assumptions1. The polynomial function is . We are to use the Rational Zeros Theorem to find all real zeros of the function3. We are to factor the function over the real numbers using the found zeros
STEP 2
The Rational Zeros Theorem states that if a polynomial has integer coefficients, then every rational zero will have the form where is a factor of the constant term and is a factor of the leading coefficient.
In our case, the constant term is -14 and the leading coefficient is1. So, the possible rational zeros of the function are the factors of -14, which are .
STEP 3
We will now substitute each possible rational zero into the function to see if it makes the function equal to zero. If it does, then it is a zero of the function.
Let's start with .
STEP 4
Calculate the value of .
Since , is not a zero of the function.
STEP 5
Continue this process for all other possible rational zeros until you find a value that makes .Let's try .
STEP 6
Calculate the value of .
So, is a zero of the function.
STEP 7
We can now use synthetic division to divide the polynomial by (since is a zero of the function, is a factor of the function).
The synthetic division should look like this
STEP 8
The result of the synthetic division gives us a new polynomial, , which is the quotient when is divided by .We can now find the zeros of this polynomial by setting it equal to zero and solving for .
STEP 9
To solve for , we can factor the quadratic equation.
STEP 10
Setting each factor equal to zero gives the other two zeros of the function.
So, the real zeros of the function are .
STEP 11
Now that we have all the real zeros of the function, we can use them to factor the function over the real numbers.
This is the factorization of the function over the real numbers.
Was this helpful?