Adding And Simplifying Polynomials: A Step-by-Step Guide
Hey math enthusiasts! Let's dive into some algebra fun. We're going to tackle adding and simplifying polynomial expressions. Don't worry, it's not as scary as it sounds. We'll break it down step by step, making it easy to understand. So, grab your pencils and let's get started!
5.1 Adding Polynomials: Combining Like Terms
Alright, guys, let's start with adding two polynomials: $3x - 7x^2 + 4$ and $3 + 2x - x^2$. The key to adding polynomials is understanding like terms. Like terms are terms that have the same variable raised to the same power. For instance, $3x$ and $2x$ are like terms, and $-7x^2$ and $-x^2$ are also like terms. Constants (numbers without variables) are also like terms, like $4$ and $3$. Our goal is to combine these like terms.
First, let's rewrite the expression so that like terms are next to each other. It's often helpful to rewrite it, grouping terms based on the powers of x, starting with the highest power. You can also think of this as arranging the terms in descending order of their exponents (also known as the degree of the term). This helps keep things organized. So, the original expressions can be written as: $-7x^2 + 3x + 4$ and $-x^2 + 2x + 3$. Now, let's add them. We'll add the like terms separately: $-7x^2 + (-x^2)$, $3x + 2x$, and $4 + 3$. Adding the $x^2$ terms gives us $-7x^2 - x^2 = -8x^2$. Adding the $x$ terms gives us $3x + 2x = 5x$. Finally, adding the constant terms, we get $4 + 3 = 7$. Putting it all together, the sum of the polynomials $3x - 7x^2 + 4$ and $3 + 2x - x^2$ is $-8x^2 + 5x + 7$. So, that's it! We've successfully added two polynomials. It's all about identifying the like terms and combining them. Remember to pay close attention to the signs (+ or -) in front of each term. Keep practicing, and you'll become a pro at this in no time. The concept of like terms is a fundamental building block in algebra, so making sure you have a solid grasp of this will help immensely as you move forward. This process might seem easy, but the fundamentals are the most important part of the learning process. You've got this!
5.2 Simplifying Algebraic Expressions
Now, let's move on to simplifying algebraic expressions. Simplifying involves applying algebraic rules to reduce an expression to its simplest form. This often includes using the distributive property, combining like terms, and performing operations like multiplication and division. Let's break down each simplification problem step by step.
5.2.1 Simplifying with the Distributive Property and Combining Like Terms
For this problem, we're going to simplify $2x(1 - x + y) - x(y - 3 + 2x)$. This problem combines the use of the distributive property and the combining of like terms. The distributive property tells us that we can multiply a term outside the parentheses by each term inside the parentheses. In this case, we have $2x$ outside the first set of parentheses and $-x$ outside the second set. First, let's distribute the $2x$: $2x * 1 = 2x$, $2x * -x = -2x^2$, and $2x * y = 2xy$. So, the first part of our expression becomes $2x - 2x^2 + 2xy$. Now, let's distribute the $-x$: $-x * y = -xy$, $-x * -3 = 3x$, and $-x * 2x = -2x^2$. So, the second part of our expression becomes $-xy + 3x - 2x^2$.
Now, we have: $(2x - 2x^2 + 2xy) - (xy - 3x + 2x^2)$. However, we must be careful with the negative sign outside the second parentheses; it changes the sign of each term inside. This gives us: $2x - 2x^2 + 2xy - xy - 3x - 2x^2$. Finally, let's combine the like terms. We have $2x$ and $-3x$ as like terms, which gives us $-x$. We have $-2x^2$ and $-2x^2$ as like terms, which gives us $-4x^2$. And we have $2xy$ and $-xy$ as like terms, which gives us $xy$. Putting it all together, the simplified expression is $-4x^2 - x + xy$. See, we’ve taken a complex-looking expression and simplified it down. Isn't that neat? Remember, the key is the methodical use of the distributive property, being mindful of the signs, and patiently combining like terms. Take your time, and double-check your work to avoid making simple mistakes. You'll master this in no time! Keep practicing, and you'll find simplifying expressions becomes much easier.
5.2.2 Simplifying by Multiplying and Dividing Monomials
Let's simplify $ rac(4a2)(-3a3)}{-6a^4}$. Here, we will use the rules for multiplying and dividing exponents. First, let's focus on the numerator. We have $(4a2)(-3a3)$. Multiply the coefficients (the numbers) $4 * -3 = -12$. Then, multiply the variables. When multiplying variables with exponents, you add the exponents. So, $a^2 * a^3 = a^{2+3} = a^5$. The numerator simplifies to $-12a^5$. Now our expression is $ rac{-12a5}{-6a4}$. Next, divide the coefficients = a^1 = a$. Therefore, the simplified expression is $2a$. That wasn't so bad, right? The key here is to remember the rules of exponents: when multiplying, add the exponents; when dividing, subtract the exponents. It's also important to be comfortable with the operations of integers. A little practice, and you'll be acing these problems! Remember, consistency in your steps will help you avoid errors, so always be careful.
5.2.3 Simplifying a Rational Expression with Factoring and Division
Now, let's simplify $ rac12x^2 - 4x}{4x} - rac{10x^2 - 15x}{5x}$. This problem involves simplifying rational expressions (fractions with variables). We'll handle each fraction separately. For the first fraction, $ rac{12x^2 - 4x}{4x}$, we can factor out a $4x$ from the numerator4x}$. Now, we can cancel out the $4x$ from the numerator and the denominator, which leaves us with $3x - 1$. For the second fraction, $ rac{10x^2 - 15x}{5x}$, we can factor out a $5x$ from the numerator{5x}$. Again, we can cancel out the $5x$ from the numerator and the denominator, which leaves us with $2x - 3$. Now, we have $(3x - 1) - (2x - 3)$. Remember the negative sign in front of the second parentheses, which affects the sign of each term inside. This becomes $3x - 1 - 2x + 3$. Finally, combine the like terms: $3x - 2x = x$ and $-1 + 3 = 2$. Thus, the simplified expression is $x + 2$. You've done great work. It's important to remember to factor correctly and to be extra cautious with the signs, especially when subtracting expressions. Practice is what makes perfect, so keep it up! Take your time, and double-check your work to avoid making simple mistakes. Keep practicing, and you'll find simplifying these expressions becomes second nature. And remember, math is all about the journey, not just the destination. Keep exploring and asking questions! It is also very helpful to organize your steps. This will allow you to see where you might be making a mistake, and it is a good study habit.
