<< Chapter < Page Chapter >> Page >

Finding the x -intercepts of a polynomial function by factoring

Find the x -intercepts of f ( x ) = x 6 3 x 4 + 2 x 2 .

We can attempt to factor this polynomial to find solutions for f ( x ) = 0.

x 2 3 x 4 + 2 x 2 = 0 Factor out the greatest common factor . x 2 ( x 4 3 x 2 + 2 ) = 0 Factor the trinomial . x 2 ( x 2 1 ) ( x 2 2 ) = 0 Set each factor equal to zero .
( x 2 1 ) = 0 ( x 2 2 ) = 0 x 2 = 0 or x 2 = 1 or x 2 = 2 x = 0 x = ±1 x = ± 2

This gives us five x -intercepts: ( 0 , 0 ) , ( 1 , 0 ) , ( −1 , 0 ) , ( 2 , 0 ) , and ( 2 , 0 ) . See [link] . We can see that this is an even function because it is symmetric about the y -axis.

Four graphs where the first graph is of an even-degree polynomial, the second graph is of an absolute function, the third graph is an odd-degree polynomial, and the fourth graph is a disjoint function.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding the x -intercepts of a polynomial function by factoring

Find the x -intercepts of f ( x ) = x 3 5 x 2 x + 5.

Find solutions for f ( x ) = 0 by factoring.

x 3 5 x 2 x + 5 = 0 Factor by grouping . x 2 ( x 5 ) ( x 5 ) = 0 Factor out the common factor . ( x 2 1 ) ( x 5 ) = 0 Factor the difference of squares . ( x + 1 ) ( x 1 ) ( x 5 ) = 0 Set each factor equal to zero .
x + 1 = 0 or x 1 = 0 or x 5 = 0 x = −1 x = 1 x = 5

There are three x -intercepts: ( −1 , 0 ) , ( 1 , 0 ) , and ( 5 , 0 ) . See [link] .

Graph of f(x)=x^6-3x^4+2x^2 with its five intercepts, (-sqrt(2), 0), (-1, 0), (0, 0), (1, 0), and (sqrt(2), 0).
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding the y - and x -intercepts of a polynomial in factored form

Find the y - and x -intercepts of g ( x ) = ( x 2 ) 2 ( 2 x + 3 ) .

The y -intercept can be found by evaluating g ( 0 ) .

g ( 0 ) = ( 0 2 ) 2 ( 2 ( 0 ) + 3 ) = 12

So the y -intercept is ( 0 , 12 ) .

The x -intercepts can be found by solving g ( x ) = 0.

( x 2 ) 2 ( 2 x + 3 ) = 0
( x 2 ) 2 = 0 ( 2 x + 3 ) = 0 x 2 = 0 or x = 3 2 x = 2

So the x -intercepts are ( 2 , 0 ) and ( 3 2 , 0 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding the x -intercepts of a polynomial function using a graph

Find the x -intercepts of h ( x ) = x 3 + 4 x 2 + x 6.

This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.

Looking at the graph of this function, as shown in [link] , it appears that there are x -intercepts at x = −3 , −2 , and 1.

Graph of g(x)=(x-2)^2(2x+3) with its two x-intercepts (2, 0) and (-3/2, 0) and its y-intercept (0, 12).

We can check whether these are correct by substituting these values for x and verifying that

h ( 3 ) = h ( 2 ) = h ( 1 ) = 0

Since h ( x ) = x 3 + 4 x 2 + x 6 , we have:

h ( −3 ) = ( −3 ) 3 + 4 ( −3 ) 2 + ( −3 ) 6 = −27 + 36 3 6 = 0 h ( −2 ) = ( −2 ) 3 + 4 ( −2 ) 2 + ( −2 ) 6 = −8 + 16 2 6 = 0 h ( 1 ) = ( 1 ) 3 + 4 ( 1 ) 2 + ( 1 ) 6 = 1 + 4 + 1 6 = 0

Each x -intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.

h ( x ) = x 3 + 4 x 2 + x 6 = ( x + 3 ) ( x + 2 ) ( x 1 )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the y - and x -intercepts of the function f ( x ) = x 4 19 x 2 + 30 x .

y -intercept ( 0 , 0 ) ; x -intercepts ( 0 , 0 ) , ( 5 , 0 ) , ( 2 , 0 ) , and ( 3 , 0 )

Got questions? Get instant answers now!

Identifying zeros and their multiplicities

Graphs behave differently at various x -intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and "bounce" off.

Suppose, for example, we graph the function shown.

f ( x ) = ( x + 3 ) ( x 2 ) 2 ( x + 1 ) 3

Notice in [link] that the behavior of the function at each of the x -intercepts is different.

Graph of h(x)=x^3+4x^2+x-6.
Identifying the behavior of the graph at an x -intercept by examining the multiplicity of the zero.

The x -intercept x = −3 is the solution of equation ( x + 3 ) = 0. The graph passes directly through the x -intercept at x = −3. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line—it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.

Questions & Answers

how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
Practice Key Terms 4

Get the best College algebra course in your pocket!





Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask