Expand (x-10)^2: A Simple Trinomial Solution

When we talk about expanding algebraic expressions, one of the most common tasks is to take a squared binomial and rewrite it as a trinomial. A trinomial is simply an expression with three terms. So, if you've been asked to express $(x-10)^2$ as a trinomial in standard form, you're essentially being asked to multiply $(x-10)$ by itself and then arrange the resulting terms in a specific order. The standard form for a polynomial means arranging the terms in descending order of their exponents. For a quadratic expression like this, it will typically look like $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. Let's dive into how we can achieve this, breaking it down step-by-step so it's easy to follow.

Understanding Binomial Expansion

Before we get to our specific problem, let's quickly recap what it means to expand a squared binomial. A binomial is an expression with two terms, like $(x-10)$. When we square a binomial, we are multiplying it by itself. So, $(x-10)^2$ means $(x-10) imes (x-10)$. The most common method for expanding this is often called the FOIL method, which is an acronym for First, Outer, Inner, Last. This method helps us ensure we multiply every term in the first binomial by every term in the second binomial. Let's apply this to our expression. The 'First' terms are $x$ and $x$, so $x imes x = x^2$. The 'Outer' terms are $x$ and $-10$, so $x imes (-10) = -10x$. The 'Inner' terms are $-10$ and $x$, so $(-10) imes x = -10x$. Finally, the 'Last' terms are $-10$ and $-10$, so $(-10) imes (-10) = +100$.

Now, we combine all these results: $x^2 + (-10x) + (-10x) + 100$. Notice that we have two terms that are like terms: $-10x$ and $-10x$. We can combine these by adding their coefficients: $-10x - 10x = -20x$. So, our expanded expression becomes $x^2 - 20x + 100$. This is a trinomial because it has three terms: $x^2$, $-20x$, and $100$. It's also in standard form because the terms are arranged in descending order of their exponents (2, then 1, then 0 for the constant term).

The $(a-b)^2$ Formula

Alternatively, we can use the algebraic identity for squaring a binomial of the form $(a-b)^2$. This identity states that $(a-b)^2 = a^2 - 2ab + b^2$. This formula is a shortcut derived from the FOIL method, and it's incredibly useful for quickly expanding squared binomials. In our case, $a$ corresponds to $x$ and $b$ corresponds to $10$. So, we can substitute these values into the formula. First, we square $a$, which gives us $a^2 = x^2$. Next, we calculate $-2ab$, which is $-2 imes x imes 10 = -20x$. Finally, we square $b$, which gives us $b^2 = 10^2 = 100$. Putting it all together, we get $x^2 - 20x + 100$. This result is the same as what we obtained using the FOIL method, and it's already in standard form. Using this formula is often faster and reduces the chance of making calculation errors, especially when dealing with more complex expressions.

Why Standard Form Matters

The standard form of a polynomial is crucial in mathematics because it provides a consistent and organized way to write expressions. For a quadratic trinomial, the standard form is $ax^2 + bx + c$. This form is particularly useful when solving quadratic equations, graphing quadratic functions, and performing polynomial operations like addition and subtraction. When an expression is in standard form, it's easy to identify its coefficients ($a$, $b$, and $c$), the degree of the polynomial (which is the highest exponent, in this case, 2), and its leading term (the term with the highest exponent, $ax^2$). This organization makes it easier to compare polynomials, understand their properties, and apply various mathematical theorems and techniques. For example, when you use the quadratic formula to solve for the roots of an equation, you need the equation to be in standard form to correctly identify the values of $a$, $b$, and $c$. Similarly, when factoring a trinomial, starting with the standard form helps in applying factoring methods systematically.

Step-by-Step Expansion of $(x-10)^2$

Let's break down the process of expressing $(x-10)^2$ as a trinomial in standard form one more time, focusing on clarity and thoroughness.

1. Identify the expression: We are given $(x-10)^2$. This means we need to multiply $(x-10)$ by itself: $(x-10)(x-10)$.

2. Apply the distributive property (or FOIL): We'll distribute each term in the first binomial to each term in the second binomial.

   • Multiply the first term of the first binomial by the first term of the second binomial: $x imes x = x^2$.
   • Multiply the first term of the first binomial by the second term of the second binomial: $x imes (-10) = -10x$.
   • Multiply the second term of the first binomial by the first term of the second binomial: $(-10) imes x = -10x$.
   • Multiply the second term of the first binomial by the second term of the second binomial: $(-10) imes (-10) = +100$.

3. Combine the results: Add all the products from the previous step: $x^2 + (-10x) + (-10x) + 100$.

4. Combine like terms: The like terms are $-10x$ and $-10x$. Combine them: $-10x - 10x = -20x$.

5. Write the final trinomial in standard form: Substitute the combined like terms back into the expression: $x^2 - 20x + 100$.

This trinomial $x^2 - 20x + 100$ is in standard form because the terms are arranged in descending order of their powers: $x^2$ (power of 2), $-20x$ (power of 1), and $100$ (power of 0, a constant term).

Common Mistakes to Avoid

When expanding expressions like $(x-10)^2$, there are a few common pitfalls that students often encounter. One of the most frequent mistakes is forgetting to square the second term or incorrectly handling the middle term. For instance, some might mistakenly think $(x-10)^2$ is simply $x^2 - 10^2$, which equals $x^2 - 100$. This is incorrect because it ignores the cross-multiplication terms (the 'Outer' and 'Inner' terms in FOIL). Another common error is with the sign of the middle term. Since we are multiplying $(x-10)$ by $(x-10)$, the product of two negative numbers ($-10$ and $-10$) is positive ($+100$). However, the middle terms, $-10x$ and $-10x$, are both negative, and when combined, they result in a negative middle term ($-20x$). If the expression were $(x+10)^2$, the middle term would be positive. It's also important to ensure the final trinomial is presented in standard form, with the highest power of $x$ first, followed by progressively lower powers, ending with the constant term. Always double-check your work, especially the signs and the distribution of terms, to ensure accuracy.

Conclusion

In summary, to express $(x-10)^2$ as a trinomial in standard form, you need to expand the expression by multiplying $(x-10)$ by itself. Using either the FOIL method or the algebraic identity $(a-b)^2 = a^2 - 2ab + b^2$, we arrive at the result $x^2 - 20x + 100$. This is a trinomial because it contains three terms, and it's in standard form with terms ordered by descending powers of $x$. Mastering this fundamental skill is a key step in your journey through algebra, opening doors to understanding more complex mathematical concepts.

For further exploration on algebraic identities and polynomial manipulation, you can visit Khan Academy's comprehensive resources on algebra. They offer detailed explanations, practice exercises, and video tutorials that can solidify your understanding of these essential mathematical topics.