Sketching the Graph of a Polynomial, 7

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:

Solution:

Consider the given equation above   

Did you notice that the given equation has a common factor which is x⁵? Let's factor the above equation as follows 

and then it is factorable by the difference of two squares as follows 

If we set y = 0, then the x-intercepts of the given equation are 0, -3, and 3.

If we set x = 0, then the y-intercept of the given equation is
 

 

 

Since we now the x-intercept, then we can sketch the location or direction of a curve as follows 

If x < -3, then y = (-)³(-)(-) = (-)
If -3 < x < 0, then y = (-)³(+)(-) = (+)
If 0 < x < 3, then y = (+)³(+)(-) = (-)
If x > 3, then y = (+)³(+)(+) = (+)

Here's the graph of a polynomial:    

Photo by Math Principles in Everyday Life

 

Sketching the Graph of a Polynomial, 6

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:

Solution:

Consider the given equation above  

Did you notice that the given equation is a perfect trinomial square? Let's factor the above equation as follows

and then it is factorable by the difference of two cubes as follows

If we set y = 0, then we can solve for the value of x at each factors which are the x-intercepts.

For , the value of x is

For , the value of x is 

Since the roots of the second factor are imaginary numbers, then we cannot accept those values. Because of this, the only x-intercept is 1.

If we set x = 0, the y-intercept of the given equation is   

Since we now the x-intercept, then we can sketch the location or direction of a curve as follows   

If x < 0, then y = (-)²(+)² = (+)
If 0 < x < 1, then y = (-)²(+)² = (+)
If x > 1, then y = (+)²(+)² = (+)

Here's the graph of a polynomial:  

Photo by Math Principles in Everyday Life

  

Sketching the Graph of a Polynomial, 5

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch the graph of a polynomial:

Solution:

Consider the given equation above 

Since the given equation, is already factored, then we can get the x-intercept by setting y = 0. The x-intercept is 1.

If we set x = 0, the y-intercept of the given equation is   

Since we now the x-intercept, then we can sketch the location or direction of a curve as follows  

If x < 0, then y = (+)⁵ = (+)
If 0 < x < 1, then y = (+)⁵ = (+)
If x > 1, then y = (-)⁵ = (-)

Here's the graph of a polynomial:  

Photo by Math Principles in Everyday Life