What are the similarities and differences for sketching a polynomial written in standard form and a polynomial written in factored form?

Question

What are the similarities and differences for sketching a polynomial written in standard form and a polynomial written in factored form?

Answer 1

Annamarie,

 

In factored form, the polynomial in x tells you quickly:

 

1. Where on the entire x-axis its zero roots occur, and whether all are distinct or some are repeated.

2. That the polynomial's value at these roots is zero since one of its multiplicative factors is zero.

3. That each root can be either a turning point or a transition point.

4. If a turning point, then the polynomial is either at a local maximum and turns back down or a local minimum and turns back up as x increases.

5. If a transition point, then the polynomial's values go from negative to positive or the reverse as x increases through the zero root.  BTW, my terminology is on the positive x values. You can make adjustments for negative x roots.

 

In standard form, the polynomial in x tells you quickly but not precisely:

 

1. What the maximum power is and therefore the maximum number of zero roots, not necessarily all distinct.

2. What its limiting behaviors might be when x grows large in the positive and negative directions. It is usually dominated by the highest power term in the polynomial.  Also if this term's power is odd, then negative x values produce a negative result from applying the power which can be reversed if the coefficient is negative. If the power is even the result is positive, but will be reversed if its coefficient is negative.  Any power of a positive x-value is still positive, but can be reversed by a negative coefficient.

3. If the polynomial has a mixture of terms with different signs or odd and even powers, then it "may" have "wiggly" graphing behavior for small or moderate values of negative or positive x as different terms may partially offset each other. 

4. In Calculus, differentiation and integration operations are easier to perform since each term can be "handled" separately and then combined for the final result.

 

Note, this could be extended into imaginary roots which I chose to ignore for clarity.  But a simple polynomial equation  like (X² + 1) = 0 only has imaginary roots +-i.  More complex polynomial equations can have a mixture of real and imaginary roots.