Let Px 2X3 5x2 4x 3 a List all possible rational zeros o

Let P(x) - 2X^3 - 5x2 - 4x + 3. (a) List all possible rational zeros of P. (Enter your answers as a comma-separated list.) (b) Find the complete factorization of P. (c) Find the zeros of P. (Enter your answers as a comma separated list.) (d) Sketch the graph of P.

Solution

P(x) = 2x^3 -5x^2 -4x +3

The factors of the leading coefficient (2) are 1 2 .The factors of the constant term (3) are 1 3 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±1/2, ±3/1, ±3/2

Substitute the possible roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

If we plug these values into the polynomial P(x), we obtain P(1)=0.

Now use factor theorem : ( 2x^3 -5x^2 -4x +3)/(x+1) = 2x^2 -7x +3

2x^2 -7x +3 = 2x^2 -6x-x+3 = 2x(x-3)-1(x-3) = (2x-1)(x-3)

x= 3; x=1/2

a) Possible rational zeros: of P: x = ±1, ±1/2, ±3 , ±3/2

b) P(x) = (2x-1)(x+1)(x-3)

c) Zeros of P(x) x= 1/2, -1 , 3

d) y intercept: P(0) = 3

3rd Graph passing through x=-1 and (0,3) is the graph for P(x)