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) = 3x^4 + 2x^3- 40x^2- 26x + 13 Find the real zeros of f. Select the correct choice below and, if necessary, fill in the answer box to complete your answer. x = Use the real zeros to factor f. f(x) =

Solution

f(x) = 3x^4 +2x^3 -40x^2 -26x + 13

The factors of the leading coefficient (3) are 1, 3 .The factors of the constant term (13) are 1, 13 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±1/3, ±13/1, ±13/3

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.

P(x)/(x+1) = (  3x^4 +2x^3 -40x^2 -26x + 13 )/(x+1)

= 3x^3x^239x+13

Again use the rational root theorem:

The factors of the leading coefficient (3) are 1 3 .The factors of the constant term (13) are 1 13 . Then the Rational Roots Tests yields the following possible solutions:

Rational Roots Tests yields the following possible solutions:

±1/1, ±1/3, ±13/1, ±13/3

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

P(x), we obtain P(1/3)=0.

So, we divide (3x^3x^239x+13)/( 3x -1) = x^2 -13

A) So, the roots of polynomial are x = -1 , 1/3 , sqrt13 , -sqrt13

B) factorise form of f(x) = (x+1)(3x-1)(x^2 +13)