Polynomial Function: Finding Zeros and Write in Factored Form

Problem: Use the rational zeros theorem to find all real zeros of the polynomial function. Use the zeros to factor f over the real numbers.

f(x)=7x^3-x^2+7x-1

Since f is a polynomial function with integer coefficients use the rational zeros theorem to find the possible zeros.

The factors of the constant term, 1 are p.
p: pm 1

The factors of the leading coefficient, 7 are q.
q: pm 1, pm 7

The possible rational zeros can be found by working out all of the possible combinations of p/q.
p/q : {pm 1}/{pm 1}, {pm 1}/{pm 7}

Simplifying these combinations give p/q : 1, -1, 1/7, -{1/7}

To test if any of these potential zeros are actual zeros, evaluate the function at these values.

This can be completed quickly using the ask feature in your calculator.

Since f(1/7) is zero, 1/7 is a zero of the function. Since the function has a zero of x=1/7 then the function has a factor of x-1/7

Use long division or synthetic division to to reduce the polynomial.

Write the factor on the outside and the function on the inside of the long division symbol. Make sure both are written in descending order and to use place holders where needed.
Divide the first term of the factor into the first term of the function. Write that value on top of the long division symbol. For this example ${7x^3}/x=7x^2$
Multiply this value by all of the terms in the expression being divided by. Write the terms under the expression you are dividing into and be sure the line up the like terms.
Change the signs of all the terms you just multiplied.
Combine like terms and bring down the next set of terms.
Repeat the process over again. Divide the first term in the factor into the new first term of the function. Write that value on top of the long division symbol. For this example
Multiply this value by all of the terms in the expression being divided by. Write the terms under the expression you are dividing into and be sure the line up the like terms.
Change the signs of all the terms you just multiplied.
Combine like terms. The steps of this process (the division algorithm) are repeated until the degree of the remainder is less than the degree of expression you are dividing by.