Simplify (p-q)(x) For Polynomials

Hey math enthusiasts! Ever get stumped by polynomial expressions? Don't sweat it, guys! Today, we're diving deep into a common problem you might encounter in algebra: simplifying expressions involving the subtraction of two polynomials. Specifically, we'll be tackling the expression $(p-q)(x)$ where $p(x) = x^2 - 1$ and $q(x) = 5(x-1)$. This is a fundamental concept, and mastering it will give you a solid foundation for more complex algebraic manipulations. We'll break down the process step-by-step, making sure you understand every bit of it. We'll also discuss why understanding polynomial subtraction is crucial in various mathematical contexts, from graphing to solving equations. So, grab your calculators, sharpen your pencils, and let's get this math party started!

Understanding Polynomial Subtraction: The Core Concept

Alright, let's get down to business with this polynomial subtraction problem. When we see $(p-q)(x)$, what we're really being asked to do is subtract the entire polynomial $q(x)$ from the entire polynomial $p(x)$. Think of it like this: you have two sets of numbers represented by these functions, and you want to find the difference between them. The notation $(p-q)(x)$ is a compact way of writing $p(x) - q(x)$. So, the first crucial step is to substitute the given expressions for $p(x)$ and $q(x)$ into this difference. We have $p(x) = x^2 - 1$ and $q(x) = 5(x-1)$. Therefore, $p(x) - q(x)$ becomes $(x^2 - 1) - (5(x-1))$. Now, pay close attention to the parentheses, guys. They are super important here because they indicate that we need to subtract the entire expression for $q(x)$, not just the first term. Failing to distribute the subtraction sign correctly is a common pitfall, so let's be extra careful. The expression $(x^2 - 1) - 5(x-1)$ is the direct representation of $(p-q)(x)$. The question asks which of the given options is equivalent to this. This means we need to simplify our expression and then see which option matches. So, let's simplify $(x^2 - 1) - 5(x-1)$. First, we need to distribute the 5 into the parentheses of $q(x)$: $5(x-1) = 5 imes x - 5 imes 1 = 5x - 5$. Now, substitute this back into our expression: $(x^2 - 1) - (5x - 5)$. The next critical step is to distribute the negative sign in front of the parentheses for $q(x)$. This means we change the sign of each term inside those parentheses: $-(5x - 5) = -5x + 5$. So, our expression now looks like $x^2 - 1 - 5x + 5$. Finally, we combine like terms. The only like terms here are the constant terms: $-1 + 5 = 4$. Therefore, the simplified expression for $(p-q)(x)$ is $x^2 - 5x + 4$. Now, we need to look at the options provided and see which one matches this simplified form or an intermediate step that is equivalent. Let's examine each option:

• A. $5(x-1) - x^2 - 1$: This is equivalent to $q(x) - p(x)$, not $p(x) - q(x)$. So, this is incorrect.
• B. $(5x - 1) - (x^2 - 1)$: This expression does not correctly represent $p(x) - q(x)$. The $q(x)$ part is written as $5x-1$, which is not equal to $5(x-1)$. So, this is incorrect.
• C. $(x^2 - 1) - 5(x-1)$: This option is exactly the expression we got when we first substituted $p(x)$ and $q(x)$ into $p(x) - q(x)$. This is the unsimplified form of $(p-q)(x)$. Since the question asks for an equivalent expression, and this is the direct translation of the subtraction, it is a valid equivalent expression. We can also see that if we were to simplify this further, we'd get our $x^2 - 5x + 4$. So, this is a strong contender.
• D. $(x^2 - 1) - 5x - 1$: This option is close, but it's missing the distribution of the negative sign to the second term of $q(x)$. It should be $(x^2 - 1) - (5x - 5)$, which would lead to $-5x + 5$. This option shows $-5x - 1$, which is incorrect. So, this is incorrect.

Based on our analysis, option C is the expression that is directly equivalent to $(p-q)(x)$ before simplification. It accurately reflects the subtraction of $q(x)$ from $p(x)$. Sometimes, math questions will test your understanding of the initial setup as well as the final simplified form. In this case, option C represents the correct setup.

Step-by-Step Breakdown of the Simplification Process

To really nail this down, let's walk through the simplification of $(p-q)(x)$ again, this time focusing on the algebraic steps that lead to the simplified polynomial $x^2 - 5x + 4$. Our starting point is, as we established, $(p-q)(x) = p(x) - q(x)$. Substituting the given functions, we get $(x^2 - 1) - (5(x-1))$. The first thing we need to do is deal with the expression for $q(x)$, which is $5(x-1)$. This involves the distributive property. We multiply the 5 by each term inside the parentheses: $5 imes x = 5x$ and $5 imes (-1) = -5$. So, $5(x-1)$ simplifies to $5x - 5$. Now, our expression becomes $(x^2 - 1) - (5x - 5)$. This is a crucial step, and it's precisely what option C represents: $(x^2-1) - 5(x-1)$. This is equivalent because it shows $p(x)$ minus the original form of $q(x)$. If the question had asked for the simplified expression, we would continue. The next major hurdle is handling the subtraction of the entire $q(x)$ expression. The minus sign in front of the parentheses $(5x - 5)$ means we need to distribute that negative sign to every term inside the parentheses. So, $-(5x - 5)$ becomes $-5x$ and $-(-5)$, which equals $+5$. Our expression transforms into $x^2 - 1 - 5x + 5$. Now comes the part where we gather and combine our