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.

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.

The factors of the leading coefficient, 7 are q.

The possible rational zeros can be found by working out all of the possible combinations of p/q.

Simplifying these combinations give

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

x	f(x)	f(x)
-1	7(-1)^3-(-1)^2+7(-1)-1	-16
-1/7	7(-1/7)^3-(-1/7)^2+7(-1/7)-1	-2.041
1/7	7(1/7)^3-(1/7)^2+7(1/7)-1	0
1	7(1)^3-(1)^2+7(1)-1	12

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 then the function has a factor of

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
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.

Write the function in factored form using the results of the long division.

Factor completely.

Problem: Use the Intermediate Value Theorem to show that the following function has a zero in the given interval.  Approximate the zero to two decimal places.

Solution:

To determine if there is a zero in the interval use the Intermediate Value theorem.  To use the Intermediate Value Theorem, the function must be continuous on the interval .  The function  is a polynomial function and polynomial functions are defined and continuous for all real numbers.

Evaluate the function at the endpoints and if there is a sign change.  If there is a sign change, the Intermediate Value Theorem states there must be a zero on the interval.  To evaluate the function at the endpoints, calculate   and .

Since one endpoint gives a negative value and one endpoint gives a positive value, there must be a zero in the interval.

We can get a better approximation of the zero by trying to figure out the next decimal point. Write out all of the values to one decimal point between -2 and -1.

x	f(x)	f(x)
-2.0	9(-2)^3+9(-2)^2-9(-2)+6	-12
-1.9
-1.8
-1.7
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
-1.0	9(-1)^3+9(-1)^2-9(-1)+6	15

Fill the table.  There are functions in your calculator that make this easier.

x	f(x)	f(x)
-2.0	9(-2)^3+9(-2)^2-9(-2)+6	-12
-1.9	9(-1.9)^3+9(-1.9)^2-9(-1.9)+6	-6.141
-1.8	9(-1.8)^3+9(-1.8)^2-9(-1.8)+6	-1.128
-1.7	9(-1.7)^3+9(-1.7)^2-9(-1.7)+6	3.093
-1.6	9(-1.6)^3+9(-1.6)^2-9(-1.6)+6	6.576
-1.5	9(-1.5)^3+9(-1.5)^2-9(-1.5)+6	9.375
-1.4	9(-1.4)^3+9(-1.4)^2-9(-1.4)+6	11.544
-1.3	9(-1.3)^3+9(-1.3)^2-9(-1.3)+6	13.137
-1.2	9(-1.2)^3+9(-1.2)^2-9(-1.2)+6	14.208
-1.1	9(-1.1)^3+9(-1.1)^2-9(-1.1)+6	14.811
-1.0	9(-1)^3+9(-1)^2-9(-1)+6	15

Use the Intermediate Value Theorem again.  Look for a sign change.  Looking down the table, there is a sign change between -1.8 and -1.7.  With this information we now know the zero is between these two values.

Repeat this process again with two decimal places between -1.8 and -1.7.

x	f(x)	f(x)
-1.80	9(-1.8)^3+9(-1.8)^2-9(-1.8)+6	-1.128
-1.79	9(-1.79)^3+9(-1.79)^2-9(-1.79)+6	-.6712
-1.78	9(-1.78)^3+9(-1.78)^2-9(-1.78)+6	-.2222
-1.77	9(-1.77)^3+9(-1.77)^2-9(-1.77)+6	.219
-1.76	9(-1.76)^3+9(-1.76)^2-9(-1.76)+6	.65242
-1.75	9(-1.75)^3+9(-1.75)^2-9(-1.75)+6	1.0781
-1.74	9(-1.74)^3+9(-1.74)^2-9(-1.74)+6	1.4962
-1.73	9(-1.73)^3+9(-1.73)^2-9(-1.73)+6	1.9066
-1.72	9(-1.72)^3+9(-1.72)^2-9(-1.72)+6	2.3096
-1.71	9(-1.71)^3+9(-1.71)^2-9(-1.71)+6	2.705
-1.70	9(-1.7)^3+9(-1.7)^2-9(-1.7)+6	3.093

Use the Intermediate Value Theorem.  Look for a sign change.  Looking down the table, there is a sign change between -1.78 and -1.77.  With this information we now know the zero is between these two values and the zero to two decimal places is -1.77 since all the numbers between -1.78 and -1.77 start with -1.77.

 

Problem:  Use the remainder theorem to find the remainder when f(x) is divided by .  Then use the factor theorem to determine whether is a factor of f(x).

Solution:

The Remainder Theorem: Let f be a polynomial function.  If is divided by , then the remainder is .

The remainder theorem gives a way to find the remainder without performing the division.  The remainder when dividing by is   Calculate to find the remainder where c is the zero of the factor .

For the example above, f(x) is divided by .  The zero of the factor can be found by setting the factor equal to zero and solve.

Use and calculate .

The remainder when f(x) is divided by is -2.

Factor Theorem: Let f be a polynomial function.  Then is a factor of f(x) if and only if .

This theorem gives of the result that if is divided by and the remainder is zero, then is a factor of f.

For the example above, the remainder when f(x) is divided by is -2.  Since the remainder is not zero is not a factor.