Simplifying Equations: From Complex To Polynomial Forms

Hey guys! Let's dive into some math problems that might seem a little intimidating at first glance, but trust me, they're totally manageable. We're going to take a look at several equations and our mission is to transform them into simpler, more familiar forms: polynomial equations. And, we'll even declare the 'degree' of each resulting polynomial. Think of the degree as the highest power of the variable in the equation – it gives us a quick idea of the equation's complexity. Ready? Let's get started!

Equation 11: Unraveling the Cubed and Squared

Alright, let's tackle the first equation, the one with the cube and squares: x³ - 7x² + 5 = x(x² - 1) + 3x² - 2. Our goal here is to manipulate this equation so that everything is on one side, and we have zero on the other, ultimately resulting in a polynomial. Here's how we'll break it down:

First, let's focus on the right side of the equation. We need to expand the terms, meaning multiply them out. So, x(x² - 1) becomes x³ - x. Now, we rewrite the entire equation with this expansion: x³ - 7x² + 5 = x³ - x + 3x² - 2. Next, we want to bring all the terms to one side. To do this, we'll subtract x³ from both sides (this cancels out the x³ terms), and add x, subtract 3x², and add 2 to both sides. This gives us: -7x² + 5 + x - 3x² + 2 = 0. We can then combine like terms: -7x² and -3x² to get -10x². Also, combine the constants 5 and 2 to get 7. Now, we have: -10x² + x + 7 = 0. This is our simplified polynomial equation!

Now, let's find the degree. The highest power of 'x' in our equation is 2 (from the x² term). Therefore, the degree of this polynomial is 2. This means it's a quadratic equation – a parabola when graphed. Not too bad, right?

So, to recap, the process involved expanding the right side, rearranging terms to one side, combining like terms, and then identifying the highest power to find the degree. Remember that understanding the degree is important because it gives you an immediate insight into the behavior and shape of the equation's graph. This systematic approach is the key to mastering these types of problems.

Equation 12: Dealing with Parentheses and Products

Now let's move on to the second equation: (y - 2)(y + 5) = (2y - 1)(y + 1) + 7. This one looks a little more complex because of all the parentheses, but don't worry, we'll break it down step by step. Our approach will be similar to the previous example: expand, simplify, and then identify the degree.

First, we need to expand the products on both sides of the equation. On the left side, (y - 2)(y + 5) becomes y² + 5y - 2y - 10, which simplifies to y² + 3y - 10. On the right side, (2y - 1)(y + 1) becomes 2y² + 2y - y - 1, which simplifies to 2y² + y - 1. So, our equation now looks like this: y² + 3y - 10 = 2y² + y - 1 + 7. Simplify the right side by combining -1 and 7, to get 6. Now, the equation is: y² + 3y - 10 = 2y² + y + 6. Now, let's move all terms to the left side: subtract 2y² from both sides, subtract y from both sides and subtract 6 from both sides. We get: y² + 3y - 10 - 2y² - y - 6 = 0. Combine the like terms: y² - 2y² to get -y², and 3y - y to get 2y, -10 and -6 to get -16. So the simplified form is: -y² + 2y - 16 = 0.

Now, let's determine the degree of the resulting polynomial. The highest power of 'y' is 2 (from the y² term). Therefore, the degree of this polynomial equation is 2. Just like the previous example, it's a quadratic equation! This tells us that if we were to graph it, we'd see a parabola again, but this time it would open downwards because of the negative sign in front of the y² term. Understanding these transformations is a fundamental skill in algebra.

Remember, the expansion of products and the careful collection of like terms are crucial for simplifying these equations. Always double-check your arithmetic, because a small error can lead to a completely different result!

Equation 13: Simplifying and Identifying

Okay, let's jump into equation number 13: y² + 7 = (y - 1)² + 3y. This one might seem a bit simpler at first glance, but we still need to go through the same steps to reduce it to a polynomial equation and identify its degree. This example will further demonstrate the importance of careful expansion and simplification.

First, we need to expand (y - 1)². Remember, (y - 1)² means (y - 1)(y - 1). Expanding this gives us y² - y - y + 1, which simplifies to y² - 2y + 1. Now, we replace (y - 1)² with its expansion, and the equation becomes: y² + 7 = y² - 2y + 1 + 3y. Combining terms on the right side: -2y + 3y, becomes just y. So the equation is y² + 7 = y² + y + 1. Next, bring everything to one side. Subtract y² from both sides, subtract y from both sides, and subtract 1 from both sides. We have: y² + 7 - y² - y - 1 = 0. Simplifying by combining like terms gives us: 7 - y - 1 = 0. Finally, simplifying the constants we get -y + 6 = 0.

Now, let's determine the degree. The highest power of 'y' is 1 (from the '-y' term). Therefore, the degree of this polynomial is 1. This means we have a linear equation! Graphically, this will produce a straight line. This simplification process is a building block for solving more complex equations and understanding the relationship between algebraic expressions and their graphical representations. Always remember to look for opportunities to combine like terms as this will simplify the equation and help you find the solution faster.

Equation 14: The Final Challenge

Alright, let's wrap things up with equation number 14: (u - 1)² = (u + 1)(u + 3) + 5. This one will be very similar to the others, but it will reinforce everything we've learned. The process remains the same: expand, simplify, and identify the degree.

First, expand (u - 1)², which is (u - 1)(u - 1). This gives us u² - u - u + 1, simplifying to u² - 2u + 1. Then expand (u + 1)(u + 3), which results in u² + 3u + u + 3, or u² + 4u + 3. Now our equation looks like this: u² - 2u + 1 = u² + 4u + 3 + 5. Combine the constants on the right side: 3 + 5 to get 8. So now we have: u² - 2u + 1 = u² + 4u + 8. To simplify further, let's move everything to the left side: subtract u² from both sides, subtract 4u from both sides, and subtract 8 from both sides. This gives us: u² - 2u + 1 - u² - 4u - 8 = 0. Combine like terms: the u² terms cancel out, -2u - 4u gives us -6u, and 1 - 8 results in -7. Thus the simplified form is: -6u - 7 = 0.

Now, let's find the degree. The highest power of 'u' is 1 (from the '-6u' term). Therefore, the degree of this polynomial is 1. Again, we have a linear equation, which would form a straight line when graphed. This reinforces how seemingly complex equations can often be simplified down to manageable, recognizable forms.

By working through these examples, you've strengthened your skills in algebraic manipulation, including expanding expressions, combining like terms, and identifying the degree of a polynomial. These are fundamental skills that will serve you well in higher levels of mathematics. Keep practicing, and you'll find these simplifications become second nature. You're doing great, and keep up the fantastic work! Remember, the more you practice, the easier it gets, and you'll become more confident in your problem-solving abilities. You've got this! And always remember to double-check your work, particularly when dealing with negative signs and parentheses – small errors can significantly change the outcome. Good job!