Factoring the Polynomial Equation 12m - m^2 - 2m^3 0

In this article, we will walk through the process of factoring the polynomial equation 12m - m^2 - 2m^3 0. This step-by-step guide will cover rearranging the equation to standard form, identifying rational roots, and using synthetic division and the quadratic formula.

Rearranging the Polynomial to Standard Form

Let's start with the given equation 12m - m^2 - 2m^3 0.

Rearrange it into standard polynomial form:

-2m^3 - m^2 2m 1 0

We can simplify this equation by factoring out a -1:

2m^3 - m^2 - 2m - 1 0

Identifying Rational Roots Using the Rational Root Theorem

Next, we will use the Rational Root Theorem to find the possible rational roots. The Rational Root Theorem states that any rational root of the polynomial is a factor of the constant term divided by a factor of the leading coefficient. For 2m^3 - m^2 - 2m - 1 0, the constant term is -1 and the leading coefficient is 2.

The possible rational roots are:

pm 1, pm 1/2

Testing the Possible Rational Roots

Let's test these values by substituting them into the polynomial:

If m 1:

2(1)^3 - (1)^2 - 2(1) - 1 2 - 1 - 2 - 1 0

Since this evaluates to 0, m 1 is a root. We can use synthetic division to divide 2m^3 - m^2 - 2m - 1 by m - 1.

Synthetic Division

Perform the synthetic division:

12-1-2-1 2310 1 2310

The result of the synthetic division is 2m^2 3m 1.

Factoring the Resulting Polynomial

Next, we need to factor the polynomial 2m^2 3m 1 using the quadratic formula. The quadratic formula is:

m frac{-b pm sqrt{b^2 - 4ac}}{2a}

For 2m^2 3m 1, the coefficients are:

a 2 b 3 c 1

Substitute these values into the quadratic formula:

m frac{-3 pm sqrt{3^2 - 4 cdot 2 cdot 1}}{2 cdot 2}

m frac{-3 pm sqrt{9 - 8}}{4}

m frac{-3 pm 1}{4}

Calculate the roots:

m frac{-3 1}{4} frac{-2}{4} -frac{1}{2} m frac{-3 - 1}{4} frac{-4}{4} -1

Complete Factorization

Thus, the complete factorization of the polynomial is:

2m^2 3m 1 2m 1m - 1

Final Factorization

Putting it all together, the original equation can be factored as:

-m(2m^2 - 1) 0

Therefore, the factored form is:

m - 1(2m 1)(m 1) 0

Solutions to the Equation

The solutions to the equation 12m - m^2 - 2m^3 0 are:

m 1 m -frac{1}{2} m -1

These are the complete factorization and solutions for the polynomial equation 12m - m^2 - 2m^3 0.