4 Letapolynomial pxhavezeros x2 x2and x3 a Use the Factor Th

4. Letapolynomial p(x)havezeros x=2, x=2,and x=3.
a. Use the Factor Theorem to find a 3rd degree polynomial that has these as

zeros.

b. Suppose you wanted a 3rd degree polynomial having these zeros but also for which the leading coefficient was 5 . How would you use your answer in part a to produce such a polynomial?

Solution

The factor theorem links factors and zeros of a polynomial. The factor theorem states that a polynomial f(x) has a factor (x –k)  if and only if f(k) = 0 (i.e. k is a root)

a. 3^rd degree polynomial p(x) having x = 2, x = -2 and x = 3 as zeros is p(x) = a (x- 2) (x + 2) (x – 3) = a(x² -4) (x – 3) = a( x³ -3x² -4x + 12) = ax³ -3ax² – 4ax + 12a where a is a constant.

b. A 3^rd degree polynomial p(x) having x = 2, x = -2 and x = 3 as zeros and the leading coefficient 5 can be obtained from p(x) in part (a) above by replacing a by 5. The required polynomial is p(x) = 5x³ -15x² – 20x + 60.