X-Intercepts: Solve Polynomial Equations Easily
Hey guys! Let's dive into a super important concept in algebra: finding the points where a graph crosses the x-axis. These points are also known as the x-intercepts, roots, or zeros of a function. Today, we're going to tackle a specific problem that will help you master this skill. We'll break down the equation, factor it, and pinpoint those crucial x-intercepts. So, grab your thinking caps, and let's get started!
The Problem: Decoding the Equation
Our mission, should we choose to accept it (and we do!), is to find the x-intercepts of the graph represented by the equation:
y = (x + 2)(x^2 + 4x + 3)
Now, at first glance, this might seem a bit intimidating, but don't worry! We're going to break it down into manageable chunks. Remember, x-intercepts are the points where the graph crosses the x-axis. What's special about these points? Well, the y-coordinate is always zero. So, to find our x-intercepts, we need to figure out what values of x will make y equal to zero. This means we need to solve the equation:
(x + 2)(x^2 + 4x + 3) = 0
This equation is a polynomial equation, and solving it involves a little bit of factoring magic. Factoring is like reverse multiplication – we're trying to break down the expression into simpler parts that are multiplied together. Lucky for us, one part is already factored: (x + 2). The other part, (x^2 + 4x + 3), is a quadratic expression, which we can also factor.
Factoring the Quadratic Expression
Let's focus on the quadratic expression: x^2 + 4x + 3. To factor this, we need to find two numbers that:
- Add up to the coefficient of the x term (which is 4)
- Multiply to the constant term (which is 3)
Think about it for a moment. What two numbers fit the bill?
The numbers 1 and 3 work perfectly! 1 + 3 = 4, and 1 * 3 = 3. So, we can factor the quadratic expression as:
x^2 + 4x + 3 = (x + 1)(x + 3)
The Fully Factored Equation
Now that we've factored the quadratic expression, we can rewrite our original equation in its fully factored form:
(x + 2)(x + 1)(x + 3) = 0
This is where the magic happens! We have a product of three factors that equals zero. The Zero Product Property states that if the product of several factors is zero, then at least one of the factors must be zero. This is a crucial concept for solving equations like this. It means that we can set each factor equal to zero and solve for x:
- x + 2 = 0
- x + 1 = 0
- x + 3 = 0
Solving for x: Unveiling the Roots
Now we have three simple linear equations to solve. Let's tackle them one by one:
-
x + 2 = 0
Subtract 2 from both sides: x = -2
-
x + 1 = 0
Subtract 1 from both sides: x = -1
-
x + 3 = 0
Subtract 3 from both sides: x = -3
So, we've found three values of x that make the equation equal to zero: x = -2, x = -1, and x = -3. These are the x-coordinates of our x-intercepts!
Identifying the Correct X-Intercept: Matching the Solution
Remember, the original question asked us to identify a point where the graph crosses the x-axis. We know that these points have the form (x, 0), and we've just found the possible x-values. Let's look back at the answer choices:
- A. (-3, 0)
- B. (2, 0)
- C. (1, 0)
- D. (3, 0)
We found that x = -3 is one of our solutions. Therefore, the point (-3, 0) is an x-intercept of the graph. Option A matches our solution!
Why Other Options Are Incorrect
It's always a good idea to understand why the other options are incorrect. This helps solidify your understanding of the concept.
-
(2, 0): If we substitute x = 2 into the original equation, we get:
y = (2 + 2)(2^2 + 4(2) + 3) = (4)(4 + 8 + 3) = (4)(15) = 60
Since y ≠0, (2, 0) is not an x-intercept.
-
(1, 0): Substituting x = 1 into the original equation:
y = (1 + 2)(1^2 + 4(1) + 3) = (3)(1 + 4 + 3) = (3)(8) = 24
Again, y ≠0, so (1, 0) is not an x-intercept.
-
(3, 0): Substituting x = 3 into the original equation:
y = (3 + 2)(3^2 + 4(3) + 3) = (5)(9 + 12 + 3) = (5)(24) = 120
Once more, y ≠0, so (3, 0) is not an x-intercept.
Key Takeaways: Mastering X-Intercepts
Okay, guys, let's recap what we've learned. Finding x-intercepts is all about setting the function equal to zero and solving for x. This often involves factoring the equation, which is a super important skill in algebra. Remember the Zero Product Property – it's your best friend when dealing with factored equations. And finally, always double-check your answers by substituting them back into the original equation to make sure they work!
To really nail this concept, let's summarize the key steps:
- Set y = 0: Replace y in the equation with 0.
- Factor the equation: Break down the expression into its simplest factors. Look for common factors and use factoring techniques for quadratic expressions.
- Apply the Zero Product Property: Set each factor equal to zero.
- Solve for x: Solve each resulting equation to find the x-coordinates of the x-intercepts.
- Write the x-intercepts as points: Express your solutions as coordinate pairs (x, 0).
- Verify your solutions: Substitute each x-value back into the original equation to ensure it equals zero.
By following these steps, you'll be able to confidently tackle any problem that involves finding x-intercepts. You've got this!
Practice Makes Perfect: Putting Your Skills to the Test
Now that we've walked through the solution, it's time to put your newfound skills to the test! Here are a few practice problems you can try:
- Find the x-intercepts of the graph of
y = (x - 1)(x^2 - 5x + 6). - What are the x-intercepts of
y = x^3 + 2x^2 - 3x? - Determine the points where the graph of
y = (2x + 1)(x^2 - 9)crosses the x-axis.
Working through these problems will help you solidify your understanding of the concepts and build your problem-solving skills. Remember to break down each problem step-by-step, factor carefully, and double-check your answers.
By understanding how to find x-intercepts, you're not just solving equations; you're gaining a deeper understanding of the relationship between algebraic equations and their graphical representations. This is a fundamental concept in mathematics that will serve you well in more advanced topics. Keep practicing, and you'll become a master of x-intercepts in no time!
So, keep practicing, and you'll be a pro at finding x-intercepts in no time! Good luck, and happy solving!
