To use Rational Zeros Theorem, express a polynomial in descending order of its exponents (starting with the biggest exponent and working to the smallest), and then take the constant term (here that's 6) and the coefficient of the leading exponent (here that's 4) and express their factors:

Constant: 6 has as factors 1, 2, 3, and 6

Coefficient: 4 has as factors 1, 2, and 4

Then all possible rational zeros must be formed by dividing a factor from the Constant list by a factor from the Coefficient list (and note that the results should be considered as both positive and negative values).

Here if you then look at the answer choices, note that since the divisor - which must be a factor from the Coefficient list of 1, 2, and 4 - cannot then be a multiple of 3, you know that  is not a possible zero of this polynomial

To apply Rational Zero Theorem, first organize a polynomial in descending order of its exponents.  Then take the constant term and the coefficient of the highest-valued exponent and list their factors:

Constant: 2 has factors of 1 and 2

Coefficient: 2 has factors of 1 and 2

Then the potential rational zeros need to be formed by dividing a factor from the Constant list by a factor from the Coefficient list. Here you only have 1s and 2s, so your options are 1, 2, and 1/2 (and note that both positive and negative values will work). There is no way to get to 1/4, meaning that that is the correct answer.

is a parabola, because of the general  structure.  The parabola opens downward because .  

Solving tells the x-value of the x-axis intercept;

The resulting x-axis intercept is: .

Setting  tells the y-value of the y-axis intercept;

The resulting y-axis intercept is:

The -intercept is , where :

The -intercept is .

This question tests one's ability to graph a polynomial function.

Step 1: Use algebraic technique to factor the function.

Separating the function into two parts...

Factoring a negative one from the second set results in...

Factoring out  from the first set results in...

The new factored form of the function is,

.

Now, recognize that the first binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

This question tests one's ability to graph a polynomial function.

Step 1: Use algebraic technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

This question tests one's ability to graph a polynomial function.

Step 1: Use algebraic technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

This question tests one's ability to graph a polynomial function.

Step 1: Use algebraic technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

This graph has zeros at 3, -2, and -4.5. This means that , , and . That last root is easier to work with if we consider it as and simplify it to . Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.

Our equation results from multiplying , which results in .

Because there are no x-intercepts, use the form , where vertex  is , so , , which gives