State the degree of the polynomial. Determine the main coefficient. List the roots of the polynomial together with their multiplicities

Question

p(x) = 7(x − 1)^2(x + 3)^3(x^2 + 2)

Answer 1

The degree of the polynomial is determined the highest exponent. 

 

The first factor has degree of 2.  Root of x=1

The second factor has degree of 3.  Root of x=-3

The last factor has degree of 2.  Root of x=±i√2

 

Add these multiplicities, we get 7.  The degree of the polynomial is 7.

 

The main coefficient is the coefficient of the leading term.  That coefficient is 7.

 

 

 

Answer 2

The 'main' coefficient would be the leading coefficient, and it's 7. The degree is also 7.

 

Since the degree is odd and the leading coefficient is positive, end behavior is:

at the far right, rising; at the far left, falling.

 

The roots are EASILY obtained by setting each factor to zero:

1 is a double root, according to the squaring of (x-1)

-3 is a triple root (multiplicity 3)coming from the repeated factor (x+3)

±i√2 are the two conjugate pure Imaginary roots.