如何进行三次多项式因式分解？求解$m^3 - 2m + 1$的分解方法

Hey there! Let's walk through exactly how to factor this cubic polynomial step by step—you'll see it's all about using a few key tricks that work for most cubics with rational roots.

Step 1: Use the Rational Root Theorem to Find a Linear Factor #

First, let's recall the Rational Root Theorem: for a polynomial like $a_nx^n + ... + a_1x + a_0$, any rational root $\frac{p}{q}$ must have $p$ as a factor of the constant term ($a_0$) and $q$ as a factor of the leading coefficient ($a_n$).

For m³ - 2m + 1:

• Constant term is 1, so possible $p$ values are ±1
• Leading coefficient is 1, so possible $q$ values are ±1
• That means our only possible rational roots are 1 and -1

Now test these values in the polynomial:

• When $m=1$: $1³ - 2(1) + 1 = 1 - 2 + 1 = 0$ → Bingo! $m=1$ is a root, so $(m - 1)$ is a factor of the polynomial.

Step 2: Factor Out the Linear Term (Two Methods) #

Now that we know $(m-1)$ is a factor, we need to find the remaining quadratic factor. Here are two easy ways to do this:

Method 1: Polynomial Long Division #

Divide m³ - 2m + 1 by $(m - 1)$:

m² + m - 1
        _______________
m - 1 | m³ + 0m² - 2m + 1
        - (m³ - m²)
        -----------
              m² - 2m
            - (m² - m)
            ---------
                  -m + 1
                - (-m + 1)
                ---------
                      0

The quotient is m² + m - 1, which can't be factored further over the integers (since its discriminant $1² - 4(1)(-1) = 5$ isn't a perfect square).

Method 2: Grouping by Rearranging Terms #

If you don't want to do long division, you can rearrange the polynomial to group terms with $(m-1)$ as a common factor:
Start with m³ - 2m + 1 → add and subtract $m²$ (this doesn't change the value, just lets us group):

m³ - m² + m² - 2m + 1

Now split into groups:

(m³ - m²) + (m² - 2m + 1)

Factor each group:

m²(m - 1) + (m - 1)²

Now factor out $(m - 1)$ from both terms:

(m - 1)(m² + (m - 1)) = (m - 1)(m² + m - 1)

Perfect—same result as the division method!

Final Result #

Putting it all together, we get:
m³ - 2m + 1 = (m - 1)(m² + m - 1)

Quick Recap for Future Cubics #

1. Use the Rational Root Theorem to test possible rational roots.
2. Once you find a root $r$, $(x - r)$ is a factor.
3. Use polynomial division or grouping to find the remaining quadratic factor.
4. Check if the quadratic can be factored further (using discriminant or trial and error).

内容的提问来源于stack exchange，提问作者kondz1v