Polynomial Functions - Pre-Calculus
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Find the point of discontinuity for the following function:
Find the point of discontinuity for the following function:
Start by factoring the numerator and denominator of the function.
A point of discontinuity occurs when a number 
is both a zero of the numerator and denominator.
Since 
is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the 
value, plug in 
into the final simplified equation.
is the point of discontinuity.
Start by factoring the numerator and denominator of the function.
A point of discontinuity occurs when a number
Since
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Factorize the following expression completely to its linear factors: 
Factorize the following expression completely to its linear factors:
Use the grouping method to factorize common terms:
Use the grouping method to factorize common terms:
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Find the point of discontinuity for the following function:
Find the point of discontinuity for the following function:
Start by factoring the numerator and denominator of the function.
A point of discontinuity occurs when a number 
is both a zero of the numerator and denominator.
Since 
is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the 
value, plug in 
into the final simplified equation.
is the point of discontinuity.
Start by factoring the numerator and denominator of the function.
A point of discontinuity occurs when a number
Since
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Find the point of discontinuity for the following function:
Find the point of discontinuity for the following function:
Start by factoring the numerator and denominator of the function.
A point of discontinuity occurs when a number 
is both a zero of the numerator and denominator.
Since 
is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the 
value, plug in 
into the final simplified equation.
is the point of discontinuity.
Start by factoring the numerator and denominator of the function.
A point of discontinuity occurs when a number
Since
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Find the vertical and horizontal asymptotes of the function
Find the vertical and horizontal asymptotes of the function
The function
is already in simplified form.
To find the vertical asymptotes, we set the denominator equal to 
and solve for 
.
yields the vertical asymptotes
To find the horizontal asymptote, we examine the largest degree of 
between the numerator and denominator
Note that
Because the largest degree of 
in the numerator is less than the largest degree of 
in the denominator, or 
we find the horizontal symptote to be
The function
is already in simplified form.
To find the vertical asymptotes, we set the denominator equal to
yields the vertical asymptotes
To find the horizontal asymptote, we examine the largest degree of
Note that
Because the largest degree of
we find the horizontal symptote to be
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Divide the polynomial 
by 
.
Divide the polynomial
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We mulitply what's below the line by 1 and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder 
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We mulitply what's below the line by 1 and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
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What is the result when 
is divided by 
?
What is the result when
Our first step is to list the coefficiens of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by 
and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficient.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with reminder 
This can be rewritten as:
Our first step is to list the coefficiens of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
This can be rewritten as:
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Divide the polynomial 
by 
.
Divide the polynomial
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by 
and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder 
This can be rewritten as:
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
This can be rewritten as:
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
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Divide the polynomial 
by 
.
Divide the polynomial
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by 
and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder 
This can be rewritten as
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
This can be rewritten as
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
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Divide the polynomial 
by 
.
Divide the polynomial
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
Remember to place a 
when there isn't a coefficient given.
We multiply what's below the line by 
and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder 
This can be rewritten as:
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
Remember to place a
We multiply what's below the line by
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
This can be rewritten as:
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
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Use synthetic division to divide 
by 
.
Use synthetic division to divide
To divide synthetically, we begin by drawing a box. On the inside separated by spaces, we write the coefficients of the terms of our polynomial being divided. On the outside, we write the root that would satisfy our binomial 
, namely 
. Leaving a space for another row of numbers, we then draw a line below our row of coefficients.
We then begin dividing by simply carrying our first coefficient (1) down below the line.
We then multiply this 1 by our divisor (3) and write the resulting product (3) below our next coefficient.
We then add the two numbers in that column and write the sum (5) below the line.
We then simply continue the process by multiplying this 5 by our divisor 3 and writing that product in the next column, adding it to the next coefficient, and continuing until we finish the columns.
We then need to translate our bottom row of numbers into the coefficients of our new quotient. Since the first column originally corresponded to our cubic term, it will now correspond to the quadratic term meaning that our 1 can be translated as 
. Similarly, our second column transitions from quadratic to linear, making our 5 become 
. Finally, our third column becomes the constant term, meaning 8 simply remains the constant 8. Finally, our former constant column becomes the column for our remainder. However, since we have a 0, we have no remainder and can disregard it.
Putting all of this together gives us a final answer of
To divide synthetically, we begin by drawing a box. On the inside separated by spaces, we write the coefficients of the terms of our polynomial being divided. On the outside, we write the root that would satisfy our binomial
We then begin dividing by simply carrying our first coefficient (1) down below the line.
We then multiply this 1 by our divisor (3) and write the resulting product (3) below our next coefficient.
We then add the two numbers in that column and write the sum (5) below the line.
We then simply continue the process by multiplying this 5 by our divisor 3 and writing that product in the next column, adding it to the next coefficient, and continuing until we finish the columns.
We then need to translate our bottom row of numbers into the coefficients of our new quotient. Since the first column originally corresponded to our cubic term, it will now correspond to the quadratic term meaning that our 1 can be translated as
Putting all of this together gives us a final answer of
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Divide using synthetic division:
Divide using synthetic division:
First, set up the synthetic division problem by lining up the coefficients. There are a couple of different strategies - for this one, we will put a -7 in the top corner and add the columns.
_________________________
The first step is to bring down the first 1. Then multiply what is below the line by the -7 in the box, write it below the next coefficient, and then add the columns:
_________________________
We can interpret this answer as meaning 
First, set up the synthetic division problem by lining up the coefficients. There are a couple of different strategies - for this one, we will put a -7 in the top corner and add the columns.
_________________________
The first step is to bring down the first 1. Then multiply what is below the line by the -7 in the box, write it below the next coefficient, and then add the columns:
_________________________
We can interpret this answer as meaning
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Using synthetic division determine which of these is a factor of the polynomial 
.
Using synthetic division determine which of these is a factor of the polynomial
Synthetic division is a short cut for doing long division of polynomials and it can only be used when divifing by divisors of the form 
. The result or quoitient of such a division will either divide evenly or have a remainder. If there is no remainder, then the "
" is said to be a factor of the polynomial. The polynomial must be in standard form (descending degree) and if a degree is skipped such as 
it must be accounted for by a "place holder".
![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/368604/gif.latex "\k|\overline{a\ 0\ b\ c\ d}\ \text{ }\vdots\")
___ 
__ __ __
__ __ ___ 
where 
is the remainder.
While doing the long division we add vertically and we multiply diagonally by k. The empty lines represent places we put the sums and products. Notice that after the first term in the top row there is a 0; this is the place holder. This is because the degrees in the polynomial skipped. When the new coefficients have been found always rewrite starting with one order lower than the highest degree of the original polynomial.
Use synthetic division to verify each factor of the form 
. Lets start with 
.
Two goes into 6 three times resulting in:
_____________________
From here we see 
will give you a remainder of zero and is therefore a factor of the polynomial 
.
Synthetic division is a short cut for doing long division of polynomials and it can only be used when divifing by divisors of the form
___
where
While doing the long division we add vertically and we multiply diagonally by k. The empty lines represent places we put the sums and products. Notice that after the first term in the top row there is a 0; this is the place holder. This is because the degrees in the polynomial skipped. When the new coefficients have been found always rewrite starting with one order lower than the highest degree of the original polynomial.
Use synthetic division to verify each factor of the form
Two goes into 6 three times resulting in:
_____________________
From here we see
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Which of the following is the correct answer (quotient and remainder format) for the polynomial 
being divided by 
.
Which of the following is the correct answer (quotient and remainder format) for the polynomial
Recall that dividing a polynomial by 
does not always result in a pefect division (remainder of 0). Sometimes there is a remainder just like in normal division. When there is a remainder, we write the answer in a certain way.
For example
where the divisor is 
, the quotient or answer is 
, the remainder is 
, and the dividend is 
.
Even though we have variables here, this is the same as noting that 
with a remainder of 
.
And how do we check to know if we have the right answer? We multiply 
and add 3 to get 15, our dividend. The same method is used for synthetic division.
Thus, for our problem:
,
we must first multiply the divisor by the quotient using the foil method (first multiplying everything in the divisor by x and then everything in the divisor by 3)
= 
now we just add the remainder which is 1 to yield 
which matches the original dividend and is therefore our answer!
Recall that dividing a polynomial by
For example
Even though we have variables here, this is the same as noting that
And how do we check to know if we have the right answer? We multiply
Thus, for our problem:
we must first multiply the divisor by the quotient using the foil method (first multiplying everything in the divisor by x and then everything in the divisor by 3)
now we just add the remainder which is 1 to yield
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Is 
a root of 
?
Is
To determine if 
is a root of the function given, you can use synthetic division to see if it goes in evenly. To set up the division problem, set up the coefficients of the function 
and then set 1 outside. Bring down the 1 (of the coefficients. Then multiply that by the 
being divided in. Combine the result of that 
with the next coefficient 
, which is 
. Then, multiply that by 
. Combine that result 
with the next coefficient 
, which gives you 
. Multiply that by 
, which gives you 
. Combine that with the last coefficient 
, whcih gives you 
. Since this is not 
, you have a remainder, which means that 
does not go in evenly to this function and is not a root.
To determine if
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Give the 
-intercept of the graph of the function
Round to the nearest tenth, if applicable.
Give the
Round to the nearest tenth, if applicable.
The 
-intercept is 
, where 
:
The 
-intercept is 
.
The
The
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Which of the following is an accurate graph of 
?
Which of the following is an accurate graph of
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: 
Solving
The resulting x-axis intercept is:
Setting
The resulting y-axis intercept is:
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Graph the following function and identify the zeros.
Graph the following function and identify the zeros.
This question tests one's ability to graph a polynomial function.
For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
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.
For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
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
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
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.
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Graph the function and identify its roots.
Graph the function and identify its roots.
This question tests one's ability to graph a polynomial function.
For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
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.
For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
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
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.
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Which could be the equation for this graph?
Which could be the equation for this graph?
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 
.
This graph has zeros at 3, -2, and -4.5. This means that
Our equation results from multiplying
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