Multiplying Expressions: A Step-by-Step Guide To 9x(2x-10)

Let's dive into the world of algebraic expressions and tackle the multiplication of $9x(2x - 10)$. This might seem daunting at first, but with a clear understanding of the distributive property and some basic algebra, you’ll find it’s quite manageable. This comprehensive guide will break down each step, ensuring you grasp the concepts thoroughly. So, grab your pencil and paper, and let’s get started!

Understanding the Distributive Property

At the heart of multiplying expressions like $9x(2x - 10)$ lies the distributive property. This fundamental principle states that multiplying a single term by an expression inside parentheses involves multiplying the term by each element within the parentheses individually. In simpler terms, if you have an expression like $a(b + c)$, you distribute the 'a' to both 'b' and 'c', resulting in $ab + ac$.

In our case, $9x$ is the term outside the parentheses, and $(2x - 10)$ is the expression inside. We need to distribute $9x$ to both $2x$ and $-10$. This means we'll perform two separate multiplications: $9x * 2x$ and $9x * -10$. Breaking it down like this helps make the problem less intimidating and easier to solve.

Think of it like this: you're sharing the $9x$ with both terms inside the parentheses. It's like giving a gift to two people; each person gets their own part of the gift. This concept is crucial for mastering algebra and handling more complex equations and expressions later on. Remember, the distributive property is your friend when it comes to simplifying and solving algebraic problems. It allows you to break down complex expressions into smaller, more manageable parts.

Step-by-Step Multiplication of 9x(2x - 10)

Now, let's apply the distributive property to our specific expression, $9x(2x - 10)$. We'll go through each step meticulously to ensure clarity. This step-by-step approach will not only help you solve this particular problem but also equip you with the skills to tackle similar algebraic expressions in the future.

Step 1: Distribute 9x to 2x

The first part of the distribution involves multiplying $9x$ by $2x$. When multiplying terms with variables, we multiply the coefficients (the numbers in front of the variables) and add the exponents of the variables. In this case, we have $9 * 2 = 18$ for the coefficients. For the variables, we have $x * x$, which is $x^2$ (since $x$ is the same as $x^1$, and $1 + 1 = 2$). Therefore, $9x * 2x = 18x^2$.

Step 2: Distribute 9x to -10

Next, we multiply $9x$ by $-10$. This is a simpler multiplication. We multiply the coefficients: $9 * -10 = -90$. The variable $x$ remains as it is since there's no other $x$ to multiply with. So, $9x * -10 = -90x$.

Step 3: Combine the Results

Now that we've distributed $9x$ to both terms inside the parentheses, we combine the results. We have $18x^2$ from Step 1 and $-90x$ from Step 2. Combining these gives us the final expression: $18x^2 - 90x$. This is the simplified form of the original expression, $9x(2x - 10)$.

By following these steps, you can confidently multiply similar expressions. The key is to break down the problem into smaller, manageable steps, applying the distributive property correctly, and paying attention to the signs and exponents. Practice makes perfect, so try working through other examples to solidify your understanding.

Simplifying the Result

After multiplying the expression $9x(2x - 10)$ and obtaining $18x^2 - 90x$, it’s always a good practice to check if the resulting expression can be further simplified. Simplification not only makes the expression cleaner but also makes it easier to work with in subsequent calculations or problem-solving scenarios. In this case, we can identify a common factor in both terms of our expression.

Identifying the Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest number or expression that divides evenly into two or more terms. To find the GCF of $18x^2$ and $-90x$, we look for the largest number that divides both 18 and 90, as well as the highest power of $x$ that is common to both terms.

For the coefficients, the GCF of 18 and 90 is 18, since 18 divides evenly into both numbers (90 = 18 * 5). For the variables, both terms have $x$, but the lowest power of $x$ is $x^1$ (or simply $x$). Therefore, the GCF of $18x^2$ and $-90x$ is $18x$.

Factoring out the GCF

Once we've identified the GCF, we can factor it out of the expression. This involves dividing each term in the expression by the GCF and writing the GCF outside a set of parentheses, with the results of the division inside the parentheses.

Dividing $18x^2$ by $18x$ gives us $x$, since $18x^2 / 18x = x$. Dividing $-90x$ by $18x$ gives us $-5$, since $-90x / 18x = -5$. Therefore, we can rewrite $18x^2 - 90x$ as $18x(x - 5)$.

The Simplified Expression

So, the simplified form of $18x^2 - 90x$ is $18x(x - 5)$. This is equivalent to the original expression but is written in a more compact and factored form. Factoring out the GCF not only simplifies the expression but can also be useful in solving equations or further manipulating algebraic expressions. Always remember to look for opportunities to simplify your results, as it can make a significant difference in the complexity of the problem.

Common Mistakes to Avoid

When multiplying expressions and simplifying them, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer. Let's discuss some of these common errors and how to steer clear of them.

Forgetting to Distribute to All Terms

One of the most frequent mistakes is not distributing the term outside the parentheses to every term inside. Remember, the distributive property requires you to multiply the term outside the parentheses by each term inside. For instance, in the expression $9x(2x - 10)$, you must multiply $9x$ by both $2x$ and $-10$. Forgetting to multiply by even one term will lead to an incorrect result. To avoid this, make sure to draw arrows connecting the term outside the parentheses to each term inside, as a visual reminder.

Incorrectly Multiplying Variables

Another common error involves the multiplication of variables. When multiplying variables with exponents, you add the exponents, not multiply them. For example, $x * x$ is $x^2$ (since $x^1 * x^1 = x^(1+1) = x^2$), not $x$. Similarly, when multiplying coefficients, remember to multiply the numbers correctly. A simple arithmetic error can throw off the entire calculation. Double-checking your multiplication, especially when dealing with larger numbers or negative signs, is always a good idea.

Sign Errors

Sign errors are also quite common, especially when dealing with negative numbers. Pay close attention to the signs when distributing. For example, in the expression $9x(2x - 10)$, multiplying $9x$ by $-10$ results in $-90x$, not $90x$. A misplaced negative sign can completely change the outcome of the problem. To minimize sign errors, it can be helpful to rewrite the expression, explicitly showing the addition or subtraction of negative terms.

Not Simplifying Completely

Finally, a common mistake is not simplifying the expression completely after performing the multiplication. This often involves not factoring out the greatest common factor (GCF). Always look for opportunities to simplify your expression further by identifying and factoring out the GCF. This not only presents the answer in its simplest form but also makes it easier to work with in subsequent steps or problems. Make it a habit to check for simplification possibilities after every calculation.

By being mindful of these common mistakes and taking the time to double-check your work, you can significantly improve your accuracy and confidence in multiplying and simplifying algebraic expressions.

Practice Problems

To solidify your understanding of multiplying expressions, working through practice problems is essential. The more you practice, the more comfortable and confident you'll become with the process. Here are a few practice problems similar to the one we've discussed, along with brief explanations to guide you.

1. Multiply and simplify: $5y(3y + 7)$

   • First, distribute $5y$ to both terms inside the parentheses: $5y * 3y$ and $5y * 7$.
   • $5y * 3y = 15y^2$ (multiply the coefficients and add the exponents of $y$).
   • $5y * 7 = 35y$ (multiply the coefficient 5 by 7).
   • Combine the results: $15y^2 + 35y$.
   • Check for further simplification: The GCF of $15y^2$ and $35y$ is $5y$. Factoring out $5y$ gives us $5y(3y + 7)$.
   • Final simplified expression: $5y(3y + 7)$.

2. Multiply and simplify: $-2a(4a - 9)$

   • Distribute $-2a$ to both terms: $-2a * 4a$ and $-2a * -9$.
   • $-2a * 4a = -8a^2$ (remember to include the negative sign).
   • $-2a * -9 = 18a$ (a negative times a negative is positive).
   • Combine the results: $-8a^2 + 18a$.
   • Check for further simplification: The GCF of $-8a^2$ and $18a$ is $2a$. Factoring out $2a$ gives us $2a(-4a + 9)$. You could also factor out -2a to get $-2a(4a - 9)$.
   • Final simplified expression: $2a(-4a + 9)$ or $-2a(4a - 9)$.

3. Multiply and simplify: $3x(6x - 2)$

   • Distribute $3x$ to both terms: $3x * 6x$ and $3x * -2$.
   • $3x * 6x = 18x^2$.
   • $3x * -2 = -6x$.
   • Combine the results: $18x^2 - 6x$.
   • Check for further simplification: The GCF of $18x^2$ and $-6x$ is $6x$. Factoring out $6x$ gives us $6x(3x - 1)$.
   • Final simplified expression: $6x(3x - 1)$.

These practice problems demonstrate the application of the distributive property and the importance of simplifying the result. Try working through these problems on your own, and then compare your solutions to the explanations provided. Remember, consistent practice is the key to mastering algebraic concepts. Don't hesitate to try more problems and challenge yourself with increasing complexity.

Conclusion

In conclusion, multiplying the expression $9x(2x - 10)$ involves applying the distributive property, which is a fundamental concept in algebra. By distributing the term outside the parentheses to each term inside, we can break down the problem into smaller, more manageable multiplications. We learned to multiply coefficients and variables correctly, paying close attention to signs and exponents. After performing the multiplication, simplifying the resulting expression by factoring out the greatest common factor (GCF) is crucial for obtaining the most concise form.

We also discussed common mistakes to avoid, such as forgetting to distribute to all terms, incorrectly multiplying variables, sign errors, and not simplifying completely. Being aware of these pitfalls can help you improve your accuracy and problem-solving skills. The practice problems provided offer an opportunity to apply the concepts learned and solidify your understanding.

Mastering the multiplication of expressions like $9x(2x - 10)$ is a stepping stone to more advanced algebraic concepts. The ability to confidently manipulate algebraic expressions is essential for success in mathematics and related fields. Remember, practice is key, so continue to work through similar problems and challenge yourself with new variations. With consistent effort and a clear understanding of the principles involved, you'll be well-equipped to tackle any algebraic expression that comes your way.

For further learning and practice on algebraic expressions, you can explore resources like Khan Academy's Algebra Section, which offers a wealth of lessons, examples, and exercises to help you build your skills.