Polynomial Functions - Algebra II
Card 0 of 268
Evaluate 
if 
and 
Evaluate
In problems with functions within one another, we must first solve the innermost function and then proceed outwards. Therefore, the first step is solving 
:
Now, we must find the values of 
:
Because our x term is squared in this function, both values end up being the same. Therefore, 59 is our final answer.
In problems with functions within one another, we must first solve the innermost function and then proceed outwards. Therefore, the first step is solving
Now, we must find the values of
Because our x term is squared in this function, both values end up being the same. Therefore, 59 is our final answer.
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Evaluate 
if 
and 
Evaluate
Beginning with the innermost function, we must first solve for 
:
We then take this value and plug it into 
:
This has no value in the real number plane, and the answer is therefore undefined.
Beginning with the innermost function, we must first solve for
We then take this value and plug it into
This has no value in the real number plane, and the answer is therefore undefined.
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Let 
, 
, and 
. What is 
?
Let
When solving functions within functions, we begin with the innermost function and work our way outwards. Therefore:
and
When solving functions within functions, we begin with the innermost function and work our way outwards. Therefore:
and
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Let 
, 
, and 
. What is 
?
Let
This problem relies on our knowledge of a radical expression 
equal to 
. The functions are subbed into one another in order from most inner to most outer function.
and
This problem relies on our knowledge of a radical expression
and
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and 
.
Determine 
.
Determine
Substituting -x into f(x). This has no effect on the 1st and 3rd terms. This changes the sign of the middle term.
Substituting -x into f(x). This has no effect on the 1st and 3rd terms. This changes the sign of the middle term.
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Polynomial Functions
Find the 
-intercepts for the polynomial function 
below:
Polynomial Functions
Find the
When finding the x-intercepts for a function, this is where the function crosses the x-axis, which means that 
or 
must equal zero.
So we first set 
which gives us:
Now in order to solve this equation, we must break down the polynomial using the "Factor by Grouping" Method.
To "Factor by Grouping" you must put the polynomial in standard form and then group into to pairs of binomials.
After doing this, one can see that there is a common factor in each group.
When an 
is taken out of the first pair we are left with 
and,
when a 
is taken out of the second pair we are left with 
again.
The goal is to make each the same and we now have two 
.
This is now a common factor on this side of the equation, so we can take out the common factor 
and we get ths result.
We can now find the x-intercepts by remembering that we origianilly set this all equal to 0.
In order for this product to equal zero, either the first or second parentheses needs to equal zero, so we set each equal to zero and solve.
and 
.
After taking the square root of both sides for 
you get 
.
When finding the x-intercepts for a function, this is where the function crosses the x-axis, which means that
So we first set
Now in order to solve this equation, we must break down the polynomial using the "Factor by Grouping" Method.
To "Factor by Grouping" you must put the polynomial in standard form and then group into to pairs of binomials.
After doing this, one can see that there is a common factor in each group.
When an
when a
The goal is to make each the same and we now have two
This is now a common factor on this side of the equation, so we can take out the common factor
We can now find the x-intercepts by remembering that we origianilly set this all equal to 0.
In order for this product to equal zero, either the first or second parentheses needs to equal zero, so we set each equal to zero and solve.
After taking the square root of both sides for
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If 
, find 
.
If
Substitute 5y in for every x:
.
Simplify:
Square the first term: 
Distribute the coefficients: 
Substitute 5y in for every x:
Simplify:
Square the first term:
Distribute the coefficients:
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If 
, find 
.
If
Substitute 
for 
in the original equation:
.
Use FOIL or the Square of a Binomial Rule to find 
.
Recall that FOIL stands for the multiplication between the First components in both binomials followed by the Outer components, then the Inner components, and lastly the Last components.
Then, Distribute: 
.
Combine like terms to simplify:
Substitute
Use FOIL or the Square of a Binomial Rule to find
Recall that FOIL stands for the multiplication between the First components in both binomials followed by the Outer components, then the Inner components, and lastly the Last components.
Then, Distribute:
Combine like terms to simplify:
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If 
, find 
.
If
To find 
, substitute 
for 
in the original equation:
.
Use FOIL or the Square of Binomial Rule to find 
.
Recall that FOIL stands for the multiplication between the First components in both binomials followed by the Outer components, then the Inner components, and lastly the Last components.
Therefore, 
.
You can then simplify the equation.
Distribute the multiplier: 
Combine like terms: 
.
To find 
, distribute 3 throughout the equation to get:
.
Subtract the two expressions:
To find
Use FOIL or the Square of Binomial Rule to find
Recall that FOIL stands for the multiplication between the First components in both binomials followed by the Outer components, then the Inner components, and lastly the Last components.
Therefore,
You can then simplify the equation.
Distribute the multiplier:
Combine like terms:
To find
Subtract the two expressions:
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Find 
if 
Find
For 
, substitue
for 
:
.
Use FOIL or square of a binomial to find 
.
Recall that FOIL stands for the multiplication between the First components in both binomials followed by the Outer components, then the Inner components, and lastly the Last components.
Therefore, 
Distribute and combine like terms to simplify:
.
For 
, first substitute 
for 
:
.
Multiply the entire expression by 3:
.
Add both expressions:
For
Use FOIL or square of a binomial to find
Recall that FOIL stands for the multiplication between the First components in both binomials followed by the Outer components, then the Inner components, and lastly the Last components.
Therefore,
Distribute and combine like terms to simplify:
For
Multiply the entire expression by 3:
Add both expressions:
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If 
, what is 
?
If
To solve this problem, plug in 2p for x in the function: 
. Then, simplify: 
.
To solve this problem, plug in 2p for x in the function:
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Let 
What is 
?
Let
What is
The question asks us to put the expression of 
into the expression for 
anyplace there is an 
:
The 2nd power needs to be distributed to both the 
and 
. The first term then becomes:
The final answer is then 
The question asks us to put the expression of
The 2nd power needs to be distributed to both the
The final answer is then
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If
,
what is
?
If
what is
To solve this problem, simply plug in 1 wherever you see x.
.
Therefore,
.
To solve this problem, simply plug in 1 wherever you see x.
Therefore,
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What are the roots of 
?
What are the roots of
In order to find the roots, we must factor the equation.
The factors of this equation are 
and 
.
Setting those two equal to zero, we get 
and 
.
In order to find the roots, we must factor the equation.
The factors of this equation are
Setting those two equal to zero, we get
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Find the product:
Find the product:
Using the FOIL (first, outer, inner, last) method, you can expand the polynomial to get
first: 
outer: 
inner: 
lasts: 
From here, combine the like terms.
Using the FOIL (first, outer, inner, last) method, you can expand the polynomial to get
first:
outer:
inner:
lasts:
From here, combine the like terms.
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Determine a possible zero: 
Determine a possible zero:
Rewrite this equation in order of high to lower powers.
Factor out an x-term from the equation. The equation becomes:
Factorize the term inside the parentheses.
Set each individual term equal to zero and solve for 
.
The zeros are: 
One of the possible root is: 
Rewrite this equation in order of high to lower powers.
Factor out an x-term from the equation. The equation becomes:
Factorize the term inside the parentheses.
Set each individual term equal to zero and solve for
The zeros are:
One of the possible root is:
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To determine the end behavior for a function, we must look at the degree and the sign associated with the function. For this function:
The degree is 6 (the highest power) which is an even number, and the sign is positive (the sign associated with the leading coefficient, which in this case is positive 1).
Taken together, we can see that we have an "even positive" function. Even functions always go the same direction at both extreme ends. Think of a parabola, both sides either both go up or both go down - this is the same for all even functions.
Since this is an even positive; both sides will approach positive infinity.
We express this mathematically when we say that as x approaches negative infinity (left side) the function will approach positive infinity:
....and as x approaches positive infinity (right side) the function again approaches positive infinity.
To determine the end behavior for a function, we must look at the degree and the sign associated with the function. For this function:
The degree is 6 (the highest power) which is an even number, and the sign is positive (the sign associated with the leading coefficient, which in this case is positive 1).
Taken together, we can see that we have an "even positive" function. Even functions always go the same direction at both extreme ends. Think of a parabola, both sides either both go up or both go down - this is the same for all even functions.
Since this is an even positive; both sides will approach positive infinity.
We express this mathematically when we say that as x approaches negative infinity (left side) the function will approach positive infinity:
....and as x approaches positive infinity (right side) the function again approaches positive infinity.
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Where is the x-value of the vertex point located for 
?
Where is the x-value of the vertex point located for
The vertex is the minimum or maximum of a parabola.
Write the vertex formula for the polynomial 
.
Substitute the coefficients.
The answer is: 
The vertex is the minimum or maximum of a parabola.
Write the vertex formula for the polynomial
Substitute the coefficients.
The answer is:
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Factorize: 
Factorize:
In order to factorize this quadratic, we will need to identify the roots of the first and last term and order it into the two binomials.
We know that it will be in the form of:
The value of 
can be divided into 
, and is the only possibility to be replaced with 
and 
.
Substitute this into the binomials.
Now we need to determine 
such that it will equal to 12, and satisfy the central term of 
.
The roots of 12 that can be interchangeable are:
The only terms that are possible are 
since
.
Remember that we must have a positive ending term!
This means that 
.
Substitute the terms.
The answer is: 
In order to factorize this quadratic, we will need to identify the roots of the first and last term and order it into the two binomials.
We know that it will be in the form of:
The value of
Substitute this into the binomials.
Now we need to determine
The roots of 12 that can be interchangeable are:
The only terms that are possible are
Remember that we must have a positive ending term!
This means that
Substitute the terms.
The answer is:
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The highest- and lowest-degree terms of a polynomial of degree 8 are 
and 
, respectively; the polynomial has only integer coefficients.
True or false: By the Rational Zeroes Theorem, it is impossible for 
to be a zero of this polynomial.
The highest- and lowest-degree terms of a polynomial of degree 8 are
True or false: By the Rational Zeroes Theorem, it is impossible for
By the Rational Zeroes Theorem (RZT), if a polynomial has only integer coefficients, then any rational zero must be the positive or negative quotient of a factor of the constant and a factor of the coefficient of greatest degree. These integers are, respectively, 24, which as as its factors 1, 2, 3, 4, 6, 8, 12, and 24, and 4, which has as its factors 1, 2, and 4.
The complete set of quotients of factors of the former and factors of the latter is derived by dividing each element of 
by each element of 
. The resulting set is
,
so any rational zero must be an element of this set. 
is not an element of this set, so by the RZT, it cannot be a zero of the polynomial.
By the Rational Zeroes Theorem (RZT), if a polynomial has only integer coefficients, then any rational zero must be the positive or negative quotient of a factor of the constant and a factor of the coefficient of greatest degree. These integers are, respectively, 24, which as as its factors 1, 2, 3, 4, 6, 8, 12, and 24, and 4, which has as its factors 1, 2, and 4.
The complete set of quotients of factors of the former and factors of the latter is derived by dividing each element of
so any rational zero must be an element of this set.
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