Verifying the roots of a polynomial after you find them is really important in math, especially in Grade 12 Algebra II. Understanding roots and zeros is a key part of this subject. It's like checking your score after a game—making sure your results are correct is very important!

After you find the roots of a polynomial, which you can write as $P(x) = 0$, there are a few simple ways to check if you got them right. Let’s start by looking at the factored form of the polynomial. If you found that $r$ is a root, this means that when you plug $r$ into the polynomial, $P(r) = 0$. This is your first test.

Step-by-Step Verification:

1. Substitution: Plug each root back into the original polynomial. For example, if you found a root $r$, calculate $P(r)$.

   • If $P(r) = 0$, then $r$ is definitely a root.
   • If it doesn’t equal 0, you might have made a mistake somewhere.

2. Factoring: If you can write the polynomial in a factored form, use that to check your roots. For example, if you can write $P(x)$ as $P(x) = (x - r_1)(x - r_2)...(x - r_n)$, then your roots are $r_1, r_2, ..., r_n$.

   • Check each factor: If you evaluate $P(r)$, it should give you 0 if $r$ is truly one of the roots.

3. Synthetic Division: You can also use synthetic division. If you think $r$ is a root, divide the polynomial by $(x - r)$.

   • If you get a remainder of 0 after dividing, then $r$ is a root.
   • This method not only confirms the root but also gives you a simpler polynomial to work with.

4. Graphing: Drawing the polynomial function can help you see its roots. You can use a graphing calculator or software to plot $P(x)$.

   • The places where the graph crosses the x-axis are the roots of the polynomial.
   • If your roots match these points, you can feel confident that you got them right.

5. Descarte’s Rule of Signs: You can use this rule as a simple check to guess how many positive and negative roots there might be. It doesn’t confirm each root, but it helps you understand possible values and avoid wrong guesses.

Practical Example:

Let’s take a polynomial like $P(x) = x^2 - 5x + 6$. You can find the possible roots using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Where $a = 1$, $b = -5$, and $c = 6$. This gives you roots of $x = 2$ and $x = 3$.

Now, let's check:

• Substitute $2$: $P(2) = 2^2 - 5*2 + 6 = 4 - 10 + 6 = 0$

• Substitute $3$: $P(3) = 3^2 - 5*3 + 6 = 9 - 15 + 6 = 0$

Since both calculations give you 0, your roots $x = 2$ and $x = 3$ are confirmed!

Conclusion:

Checking the roots of a polynomial is an important step that helps you understand and trust what you’ve learned in algebra. Whether you use substitution, factoring, synthetic division, graphing, or Descarte’s Rule, each method has its own benefits. Practicing these techniques not only strengthens your understanding, but it also gets you ready for more complex math concepts. Remember, learning about polynomials and their roots is like getting better at a game: practice makes perfect, and verifying your work ensures success!