Use the rational zeros theorem to find all the real zeros of

Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x) = x^3 + 3x^2 - 33x - 35 Find the real zeros of f. Select the correct choice below and, if necessary, fill in the answer bex to complete your answer.

Solution

f(x) = x^3 +3x^2 -33x -35

Use rational zero theorem:

The factor of the leading coefficient (1) is 1 .The factors of the constant term (-35) are 1 5 7 35 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±5/1, ±7/1, ±35/1

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.

So, now we use factor theorem: divide (x^3 +3x^2 -33x -35)/(x+1) = x^2 +2x -35

now solve the quadratic equation : x^2 +2x -35 = x^2 +7x -5x -35

= x(x+7) -5(x+7) = (x-5)(x+7) =0

x= 5, -7

So, rational zeros of f(x) = -1, 5, -7