Sketching the Graph of a Polynomial, 9

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 trinomial inside the parenthesis is factorable by the product of two binomials? Let's factor the above equation as follows

If we set y = 0, then the x-intercepts of the given equation are 3, and -1.
    
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 < -1, then y = (-)²(-)² = (+)
If -1 < x < 0, then y = (-)²(+)² = (+)
If 0 < x < 3, then y = (-)²(+)² = (+)
If x > 3, then y = (+)²(+)² = (+)

Here's the graph of a polynomial:   

Photo by Math Principles in Everyday Life

 

Sketching the Graph of a Polynomial, 8

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 can be factored by grouping? 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 2.
   
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(+) = (+)
If 0 < x < 2, then y = (-)²(+) = (+)
If x > 2, then y = (+)²(+) = (+)

Here's the graph of a polynomial:   

Photo by Math Principles in Everyday Life

 
 

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