College algebra

Question

Find all rational zeroes of the polynomial and write in complete factored form: f(x) = x^3+3x^2-4

Answer 1

f(x) = x³ + 3x² - 4

 

f(1) = 1 + 3 - 4 = 0

 

So, x - 1 is a factor of f(x).

 

divide f(x) by x - 1:      1⌋    1        3       0         -4

                                                          1       4           4                               

                                                1        4       4          0

 

f(x) = (x - 1)(x² + 4x + 4)

 

        = (x - 1)(x + 2)(x + 2)    

 

 

Answer 2

f(x) = x³ + 3x² - 4

 

f(1) = 1 + 3 - 4 = 0 ⇒  1 is one zero of the function above.

f(-1)= -1 + 3 - 4 = -2 ⇒ -1 is not a zero of the function f.

f(-2) = -8 + 12 - 4 = 0  -2 is another zero of f.

 

f(x) = (x - 1)[x - (-2)](x + p) ; all we need to do is determine the parameter p to solve our problem.

 

(x-1)(x+2)(x+p) = x³ + 3x² - 4

(x² + x - 2)(x+p) = x³ + 3x² + 0x - 4

x³ + (p+1)x² + (p-2)x - 2p = x³ + 3x² + 0x - 4 ⇒ p = 2.

 

f(x) = (x -1)(x+2)(x+2)

 

Notice that -2 is twice a zero of this function. The notion of multiplicity is involved ... and is not part of the question asked!

Answer 3

By the Rational Root Theorem, the possible rational roots are ±1, ±2, and ±4.

 

Using synthetic substitution 1 is a root (zero).

 

One factor is (x - 1). The other is x² + 4x +4.

You can factor to determine the other two zeros.