Factor the polynomial completely Px x3 x2 4x 4 Px Find

Factor the polynomial completely. P(x) = x^3 + x^2 + 4x + 4 P(x) = _____ Find all its zeros. State the multiplicity of each zero. (Order your answers from smallest to largest real, followed by complex answers ordered smallest to largest real part, then smallest to largest imaginary part.) x = _______ with multiplicity ________ x = _______ with multiplicity ________ x = _______ with multiplicity ________

Solution

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

The factor of the leading coefficient (1) is 1 .The factors of the constant term (4) are 1 2 2 4 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±2/1, ±2/1, ±4/1

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.

To find remaining zeros we use Factor Theorem. This theorem states that if pq is root of the polynomial then this polynomial can be divided with qxp. In this example:

( x^3 + x^2 + 4x +4)/(x +1) = x^2 +4

Roots of x^2 +4 ; x = 2i , -2i

P(x) = (x+1)(x -2i)(x+2i)

x = -1 with multiplicty 1

x = 2i with multiplicty 1

x = -2i with multiplicty 1