<< 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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask