Subtracting Polynomials: A Step-by-Step Guide

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Subtracting Polynomials: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of polynomials and tackling a common question: how to subtract one polynomial from another. Specifically, we'll be looking at how to subtract $x^2 + 6x - 10$ from $-8x^2 - 3x$. Don't worry, it's not as scary as it looks! We'll break it down step-by-step, making sure you understand each part of the process. So, grab your pencils and paper, and let's get started!

Understanding Polynomial Subtraction

Before we jump into the actual problem, let's quickly review what polynomials are and the general idea behind subtracting them. Polynomials are expressions made up of variables and coefficients, combined using addition, subtraction, and multiplication, where the exponents of the variables are non-negative integers. Examples of polynomials include $2x^2 + 3x - 1$, $5x - 7$, and even just the number $4$. The key here is understanding the different terms within a polynomial. A term is a single part of the polynomial, such as $2x^2$ or $-3x$. When we subtract polynomials, we're essentially combining like terms. Like terms are those that have the same variable raised to the same power. For instance, $3x^2$ and $-5x^2$ are like terms because they both have $x^2$, but $3x^2$ and $2x$ are not like terms because they have different powers of $x$.

The basic idea behind subtracting polynomials is similar to subtracting regular numbers. Think of it like this: if you have $10 and you want to subtract 3, you're essentially taking away 3 from 10. With polynomials, we're doing the same thing, but instead of just numbers, we're dealing with expressions. The key step in polynomial subtraction is distributing the negative sign. When you subtract an entire polynomial, you're subtracting each term within that polynomial. This means you need to change the sign of each term in the polynomial you're subtracting. This is the most crucial part, so pay close attention! After distributing the negative sign, we simply combine like terms to get our final answer. We'll walk through this process in detail with our example problem.

Step-by-Step Solution: Subtracting $x^2 + 6x - 10$ from $-8x^2 - 3x$

Now, let's tackle the main question: how to subtract $x^2 + 6x - 10$ from $-8x^2 - 3x$. We'll break this down into easy-to-follow steps.

Step 1: Write Out the Subtraction

First, we need to write out the subtraction problem correctly. It's super important to pay attention to the order because subtraction isn't commutative (meaning a - b is not the same as b - a). We're subtracting $x^2 + 6x - 10$ from $-8x^2 - 3x$, so we write it as:

(−8x2−3x)−(x2+6x−10)(-8x^2 - 3x) - (x^2 + 6x - 10)

Notice the parentheses! These are crucial because they tell us we're subtracting the entire expression $(x^2 + 6x - 10)$. If we didn't use parentheses, we might accidentally only subtract the first term, which would give us the wrong answer.

Step 2: Distribute the Negative Sign

This is the most important step, guys! We need to distribute the negative sign in front of the second set of parentheses to each term inside. Remember, subtracting a quantity is the same as adding its negative. So, we're essentially multiplying each term inside the parentheses by -1:

(−8x2−3x)−(x2+6x−10)=−8x2−3x−x2−6x+10(-8x^2 - 3x) - (x^2 + 6x - 10) = -8x^2 - 3x - x^2 - 6x + 10

Notice how the signs of each term inside the second set of parentheses have changed:

  • x^2$ became $-x^2

  • +6x$ became $-6x

  • -10$ became $+10

This is why parentheses are so important! They ensure we distribute the negative sign to every term in the polynomial we're subtracting. This step is the most common place where errors happen, so double-check your signs!

Step 3: Combine Like Terms

Now that we've distributed the negative sign, we can combine the like terms. Remember, like terms have the same variable raised to the same power. Let's identify the like terms in our expression:

−8x2−3x−x2−6x+10-8x^2 - 3x - x^2 - 6x + 10

  • We have two terms with $x^2$: $-8x^2$ and $-x^2$
  • We have two terms with $x$: $-3x$ and $-6x$
  • We have one constant term: $+10$

Now, let's combine them. Remember, when we combine like terms, we only add or subtract the coefficients (the numbers in front of the variables). We leave the variable and its exponent the same:

  • −8x2−x2=−9x2-8x^2 - x^2 = -9x^2

  • −3x−6x=−9x-3x - 6x = -9x

  • The constant term $+10$ stays the same.

So, combining the like terms gives us:

−9x2−9x+10-9x^2 - 9x + 10

Step 4: Write the Final Answer

We've done it! We've successfully subtracted $x^2 + 6x - 10$ from $-8x^2 - 3x$. Our final answer is:

−9x2−9x+10-9x^2 - 9x + 10

This is the simplified polynomial we get after performing the subtraction. It's important to write your answer in standard form, which means arranging the terms in descending order of their exponents. In this case, we already have the terms in standard form (the $x^2$ term first, then the $x$ term, and finally the constant term).

Example Problems for Practice

To really nail this concept, let's look at a few more example problems. Practice makes perfect, guys!

Example 1: Subtract $2y^2 - 5y + 3$ from $7y^2 + y - 8$

  1. Write out the subtraction: $(7y^2 + y - 8) - (2y^2 - 5y + 3)$
  2. Distribute the negative sign: $7y^2 + y - 8 - 2y^2 + 5y - 3$
  3. Combine like terms:

    • 7y2−2y2=5y27y^2 - 2y^2 = 5y^2

    • y+5y=6yy + 5y = 6y

    • −8−3=−11-8 - 3 = -11

  4. Final answer: $5y^2 + 6y - 11$

Example 2: Subtract $-3a^2 + 4a$ from $a^2 - 9$

  1. Write out the subtraction: $(a^2 - 9) - (-3a^2 + 4a)$
  2. Distribute the negative sign: $a^2 - 9 + 3a^2 - 4a$
  3. Combine like terms:

    • a2+3a2=4a2a^2 + 3a^2 = 4a^2

    • We only have one term with $a$, so it stays $-4a$
    • We only have one constant term, so it stays $-9$
  4. Final answer: $4a^2 - 4a - 9$

Example 3: Subtract $5b^3 - 2b + 1$ from $b^3 + 4b^2 - 6$

  1. Write out the subtraction: $(b^3 + 4b^2 - 6) - (5b^3 - 2b + 1)$
  2. Distribute the negative sign: $b^3 + 4b^2 - 6 - 5b^3 + 2b - 1$
  3. Combine like terms:

    • b3−5b3=−4b3b^3 - 5b^3 = -4b^3

    • We only have one term with $b^2$, so it stays $4b^2$
    • We only have one term with $b$, so it stays $2b$

    • −6−1=−7-6 - 1 = -7

  4. Final answer: $-4b^3 + 4b^2 + 2b - 7$

Common Mistakes to Avoid

Guys, it's super important to be aware of common mistakes so you can avoid them! Here are a few things to watch out for when subtracting polynomials:

  • Forgetting to distribute the negative sign: This is the most common mistake! Make sure you change the sign of every term inside the parentheses you're subtracting.
  • Combining non-like terms: You can only combine terms that have the same variable raised to the same power. For example, you can't combine $x^2$ and $x$.
  • Incorrectly adding/subtracting coefficients: Double-check your arithmetic when combining like terms. Make sure you're adding or subtracting the coefficients correctly.
  • Not writing the answer in standard form: While not technically incorrect, it's good practice to write your answer in standard form (descending order of exponents) so it's easier to read and compare.

Tips and Tricks for Success

Here are a few tips and tricks to help you master polynomial subtraction:

  • Write it out clearly: Especially when you're first learning, write out each step clearly and carefully. This will help you avoid mistakes.
  • Use different colors or symbols to identify like terms: This can be especially helpful for longer polynomials with many terms.
  • Double-check your work: After you've finished a problem, take a few minutes to go back and check each step. Did you distribute the negative sign correctly? Did you combine like terms properly? Is your answer in standard form?
  • Practice, practice, practice! The more you practice, the more comfortable you'll become with polynomial subtraction. Work through lots of examples, and don't be afraid to ask for help if you get stuck.

Conclusion

So, there you have it! Subtracting polynomials might seem tricky at first, but by following these steps and practicing regularly, you'll become a pro in no time. The key takeaway is to remember to distribute the negative sign carefully and then combine those like terms. Keep practicing, and you'll be subtracting polynomials like a boss! Remember, math is like a puzzle, and each step is a piece that fits together to create the solution. Keep practicing, and you'll become a master puzzle solver!