Polynomial Subtraction: Step-by-Step Guide

Hey guys! Today, we're diving into the world of polynomial subtraction. Polynomials might sound intimidating, but trust me, they're not as scary as they seem. We'll break down a few examples step by step, so you can master this skill in no time. Let's get started!

1. Evaluating the Expression $2x^2 - x + 6$

Okay, so the first question is a bit different; it's about evaluating an expression. But hey, it's still a good warm-up! The expression we're dealing with is $2x^2 - x + 6$. Now, to evaluate this, we need a value for $x$. Since the question doesn't provide one, let's consider what happens if we did have a value for $x$. Say, for instance, $x = 2$.

If $x = 2$, we'd plug it into the expression like this:

$2(2)^2 - (2) + 6 = 2(4) - 2 + 6 = 8 - 2 + 6 = 12$

So, when $x = 2$, the expression equals 12. The important thing to remember here is the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

But wait! What if we wanted to explore this expression a bit more generally? We can talk about its structure. The expression $2x^2 - x + 6$ is a quadratic expression. The $2x^2$ term is the quadratic term, $-x$ is the linear term, and $6$ is the constant term. Understanding these components helps us when we manipulate and solve equations involving these types of expressions. Also, the coefficient of the $x^2$ term (which is 2) tells us something about the parabola if we were to graph this expression – it opens upwards and is a bit "narrower" than the standard $x^2$ parabola.

So, in summary, to properly "answer" this question, you really need a value for $x$. Without a specific value, you can't simplify it to a single number, but you can understand the structure and how to evaluate it when you do have a value for $x$. Keep this in mind, guys, as you tackle similar problems!

2. Subtracting $3x^2 + 4x - 1$ from $x^2 + 1$

Now let's dive into some actual subtraction. We need to subtract $3x^2 + 4x - 1$ from $x^2 + 1$. Remember, the order matters! We're starting with $x^2 + 1$ and taking away $3x^2 + 4x - 1$. Here's how we set it up:

$(x^2 + 1) - (3x^2 + 4x - 1)$

First, distribute the negative sign to each term inside the second parenthesis:

$x^2 + 1 - 3x^2 - 4x + 1$

Now, combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have $x^2$ terms, $x$ terms, and constant terms:

$(x^2 - 3x^2) + (-4x) + (1 + 1)$

Combine the coefficients of the $x^2$ terms: $1x^2 - 3x^2 = -2x^2$. So we have:

$-2x^2 - 4x + 2$

And that's it! The result of subtracting $3x^2 + 4x - 1$ from $x^2 + 1$ is $-2x^2 - 4x + 2$. This is a quadratic expression in standard form, which means the terms are arranged in descending order of their exponents.

To make sure you really understand this, let’s think about what happens if you mess up the order. If you subtracted $(x^2 + 1)$ from $(3x^2 + 4x - 1)$, you would get a completely different result. Always pay attention to the wording of the problem! Math is not only about the steps, is also about precision.

3. Subtracting $4x^2 + 7x - 5$ from $9x^2 - 2x + 3$

Alright, let's tackle another subtraction problem. This time we're subtracting $4x^2 + 7x - 5$ from $9x^2 - 2x + 3$. Again, watch that order!

$(9x^2 - 2x + 3) - (4x^2 + 7x - 5)$

Distribute the negative sign:

$9x^2 - 2x + 3 - 4x^2 - 7x + 5$

Combine like terms:

$(9x^2 - 4x^2) + (-2x - 7x) + (3 + 5)$

Simplify:

$5x^2 - 9x + 8$

So, the result of subtracting $4x^2 + 7x - 5$ from $9x^2 - 2x + 3$ is $5x^2 - 9x + 8$. This is already in standard form. The standard form of a polynomial helps make it easier to compare with other polynomials and to perform further operations, such as factoring or finding roots.

Let's think about why standard form is useful. Imagine you're trying to add two polynomials together. If they're both in standard form, it's super easy to line up the like terms and add their coefficients. It's all about organization, guys!

4. Subtracting $5x^2 - 7x - 6$ from $9x^2 + 3x - 4$

One more subtraction problem to solidify your understanding! We're subtracting $5x^2 - 7x - 6$ from $9x^2 + 3x - 4$.

$(9x^2 + 3x - 4) - (5x^2 - 7x - 6)$

Distribute that negative sign:

$9x^2 + 3x - 4 - 5x^2 + 7x + 6$

Combine those like terms:

$(9x^2 - 5x^2) + (3x + 7x) + (-4 + 6)$

Simplify:

$4x^2 + 10x + 2$

Therefore, when we subtract $5x^2 - 7x - 6$ from $9x^2 + 3x - 4$, we get $4x^2 + 10x + 2$. This is, once again, already in standard form. You'll notice a pattern here - the result of subtracting polynomials will always be another polynomial (or zero!). Polynomials are "closed" under the operation of subtraction, which is a fancy way of saying you'll always end up with another polynomial.

Also, let's consider a real-world analogy (as real as math gets, haha!). Imagine you're managing a budget. One polynomial represents your income, and another represents your expenses. Subtracting your expenses from your income (both represented as polynomials) gives you your profit, which is also a polynomial. Cool, right?

Conclusion

So there you have it! We've covered evaluating expressions and subtracting polynomials. The key takeaways are: pay attention to the order of operations, distribute the negative sign carefully when subtracting, combine like terms, and write your answer in standard form. Keep practicing, and you'll be a polynomial subtraction pro in no time!