Master Polynomial Evaluation: Solving p(1) Step-by-Step

Evaluating polynomials is a fundamental skill in algebra, and today we’re going to break down the process using an example that’s as simple as it is illuminating. Ready to tackle the polynomial ( p(x) = x^3 - 4x^2 + x - 6 )? Let's focus on how to find ( p(1) ), step by step!

What Do We Need to Do?

First off, what does it mean to evaluate a polynomial at a specific point? Essentially, it's about substituting a number for the variable in the polynomial expression and simplifying. When we ask for ( p(1) ), we're looking for the value of the polynomial when ( x ) is replaced by 1. You know what? This method is not just handy; it’s the bread and butter of polynomial math.

Let’s Dive Into the Calculation

Let's get into the nitty-gritty; here's how we evaluate ( p(1) ):

1. Substitute 1 into the Polynomial:
   We begin by plugging 1 into our expression:
   [ p(1) = (1)^3 - 4(1)^2 + (1) - 6 ]

2. Compute Each Term:
   Now, let's break this down into manageable pieces:

• ((1)^3 = 1)
• (-4(1)^2 = -4)
• ((1) = 1)
• (-6 = -6)

3. Combine the Results:
   We’re ready to combine those calculated results into one number:
   [ p(1) = 1 - 4 + 1 - 6 ]

4. Simplify It:
   Let’s tackle this step by step:

• Start with ( 1 - 4 ) which gives us (-3).
• Then, (-3 + 1) leads us to (-2).
• Finally, we subtract 6: (-2 - 6) equals (-8).

So, voila! ( p(1) ) evaluates to (-8). Now that’s a wrap—simple, right? This technique of substitution shines a light on polynomial evaluation, making the once-daunting task feel like a walk in the park!

Why This Matters

You might wonder, why bother mastering this technique? Understanding how to evaluate polynomials at specific points not only lays a solid groundwork for more complex algebra concepts, but it also hones analytical skills that will be invaluable in future mathematical pursuits. Whether you're preparing for an exam or just curious about polynomials, getting comfortable with this skill is essential.

Wrapping It Up

The world of polynomials is all about patterns and relationships. As you practice evaluating polynomials like ( p(x) = x^3 - 4x^2 + x - 6 ), you'll not only enhance your problem-solving abilities but also develop a deeper appreciation for the elegance of algebra. Keep practicing, and soon enough, these evaluations will feel as natural as brushing your teeth in the morning.