Question

Use the factor theorem to show that x+2 is a factor of f(x). The find all real zeroes for the polynomial given that x+2 is a factor of f(x). f(x) = x^3 -5x^2 -2x +24

Answer 1

Solution :

f(x) = x³ - 5x² - 2x + 24

This is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.

The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction p/q, where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factor of the leading coefficient (1) is 1 .The factors of the constant term (24) are 1, 2, 3, 4, 6, 8, 12 ,24 . Then the Rational Roots Tests yields the following possible solutions :

± 1/1, ± 2/1, ± 3/1, ± 4/1, ± 6/1, ± 8/1, ± 12/1, ± 24/1

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( − 2 ) = 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 qx − p.

Divide P(x) with ( x + 2 )

( x³ - 5x² - 2x + 24 ) / ( x + 2 ) = x² - 7x + 12

Polynomial x²−7x+12 can be used to find the remaining roots.

x²−7x+12 is a second degree polynomial.

The roots of the polynomial x³− 5x²− 2x + 24 are :

x₁ = - 2, x₂ = 3, x₃ = 4