Hey there, math enthusiasts! Today, we're diving into the world of polynomials, specifically tackling the expression 3y² + (y + 7)² - 15. Our mission? To simplify this expression and write it in the standard form. If you've ever felt a little lost in the algebraic jungle, don't worry, we'll break it down step by step, making sure everyone can follow along. So, grab your pencils, and let's get started!
Understanding Polynomials and Standard Form
Before we jump into the simplification, let's quickly recap what polynomials are and what we mean by "standard form." Polynomials, guys, are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Think of them as the building blocks of algebra, and they come in all shapes and sizes.
The standard form of a polynomial is a specific way of writing it that makes it easy to compare and work with. In standard form, the terms are arranged in descending order of their exponents. For example, if we have a polynomial with terms like x², x, and a constant, we'd write it with the x² term first, then the x term, and finally the constant term. This organized structure helps us easily identify the degree of the polynomial (the highest exponent) and the leading coefficient (the coefficient of the term with the highest exponent).
When dealing with polynomial simplification, we aim to combine like terms. Like terms are those that have the same variable raised to the same power. For instance, 3y² and 5y² are like terms because they both have y raised to the power of 2. We can add or subtract their coefficients to combine them. However, 3y² and 5y are not like terms because the exponents are different. Understanding this concept is crucial for simplifying polynomials correctly.
Now, why is the standard form so important? Well, imagine you're comparing two complex polynomials. If they're both in standard form, it's much easier to see which one has a higher degree or which has a larger leading coefficient. This can be particularly useful when solving equations or analyzing functions. The standard form also helps in identifying patterns and making further algebraic manipulations smoother. Moreover, it’s universally accepted, which means that anyone looking at your simplified polynomial will immediately understand its structure and properties.
So, with these foundational concepts in mind, we’re well-equipped to tackle the polynomial 3y² + (y + 7)² - 15. We’ll expand, combine like terms, and arrange everything neatly in standard form. Stay tuned as we dive into the step-by-step simplification process!
Step-by-Step Simplification of 3y² + (y + 7)² - 15
Alright, let’s get our hands dirty and simplify the polynomial 3y² + (y + 7)² - 15. We'll take it one step at a time to make sure we don't miss anything. The key here is to follow the order of operations (PEMDAS/BODMAS) and be meticulous with our algebraic manipulations.
Step 1: Expanding the Squared Term
The first thing we need to do is to expand the squared term, (y + 7)². Remember, squaring a binomial means multiplying it by itself: (y + 7)² = (y + 7)(y + 7). To do this, we can use the FOIL method (First, Outer, Inner, Last) or the distributive property. Let’s use the distributive property, which is essentially the same thing but perhaps a bit clearer for some:
(y + 7)(y + 7) = y(y + 7) + 7(y + 7)
Now, we distribute y and 7 across the terms inside the parentheses:
y(y + 7) = y² + 7y 7(y + 7) = 7y + 49
So, combining these, we get:
(y + 7)² = y² + 7y + 7y + 49 = y² + 14y + 49
Step 2: Substituting the Expanded Term Back into the Original Polynomial
Now that we've expanded (y + 7)², we can substitute it back into our original polynomial:
3y² + (y + 7)² - 15 becomes 3y² + (y² + 14y + 49) - 15
Step 3: Removing Parentheses and Combining Like Terms
Next, we remove the parentheses. Since we’re adding the expression inside the parentheses, we can simply drop them:
3y² + y² + 14y + 49 - 15
Now, let’s identify and combine the like terms. We have two terms with y² (3y² and y²), one term with y (14y), and two constant terms (49 and -15). Combining them:
- y² terms: 3y² + y² = 4y²
- y term: 14y (no other like terms)
- Constant terms: 49 - 15 = 34
So, our simplified polynomial is:
4y² + 14y + 34
Step 4: Writing the Simplified Polynomial in Standard Form
Finally, we need to write our simplified polynomial in standard form. As we discussed earlier, standard form means arranging the terms in descending order of their exponents. In this case, our polynomial is already in standard form:
4y² + 14y + 34
The term with the highest exponent (y²) comes first, followed by the term with y, and then the constant term. The standard form makes it easy to see the structure and degree of the polynomial. Thus, after simplifying and arranging the terms, we have our final answer.
So, there you have it! We’ve successfully simplified the polynomial 3y² + (y + 7)² - 15 step by step. Now, let's see which of the provided options matches our result.
Matching the Simplified Polynomial with the Options
Okay, now that we've simplified the polynomial 3y² + (y + 7)² - 15 to 4y² + 14y + 34, let's match our result with the given options. This is a crucial step to ensure we've not only done the algebra correctly but also that we can identify the correct answer in a multiple-choice setting.
Let’s revisit the options:
A. 4y² + 34 B. 3y² + y + 34 C. 4y² + 14y + 34 D. 4y⁴ + 34
Now, let's compare each option with our simplified polynomial, 4y² + 14y + 34:
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Option A: 4y² + 34 This option has the 4y² term and the constant term 34, but it’s missing the 14y term. So, it’s not a match.
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Option B: 3y² + y + 34 This option has different coefficients for the y² and y terms compared to our simplified polynomial. The coefficient of y² is 3 instead of 4, and the coefficient of y is 1 instead of 14. So, this is also not a match.
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Option C: 4y² + 14y + 34 This option perfectly matches our simplified polynomial! It has the same terms and coefficients: 4y² for the squared term, 14y for the linear term, and 34 as the constant term. So, this looks like our winner.
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Option D: 4y⁴ + 34 This option has a y⁴ term, which means the degree of the polynomial is different from our result. Our simplified polynomial has a degree of 2 (the highest exponent is 2), while this option has a degree of 4. So, this is not a match.
After carefully comparing each option with our simplified polynomial, it’s clear that Option C, 4y² + 14y + 34, is the correct answer. We’ve successfully simplified the polynomial and identified the matching option. Pat yourselves on the back, guys; we nailed it!
When faced with multiple-choice questions like this, it's always a good idea to go through this process of elimination. Even if you're confident in your answer, comparing it with the other options can help you catch any small errors you might have made. This way, you can approach the problem systematically and ensure accuracy.
Common Mistakes to Avoid When Simplifying Polynomials
When simplifying polynomials, it's easy to make a few common mistakes if you're not careful. Let's talk about these pitfalls so we can avoid them in the future. Think of this as our algebra safety briefing! By being aware of these errors, we can approach polynomial simplification with confidence and accuracy.
1. Incorrectly Expanding Squared Terms:
One of the most frequent mistakes happens when expanding squared binomials like (y + 7)². Many people mistakenly think that (y + 7)² is equal to y² + 7². However, this is incorrect. Remember, (y + 7)² means (y + 7)(y + 7), and we need to use the FOIL method or the distributive property to expand it correctly. Failing to do so will lead to an incorrect simplified polynomial. Always remember the middle term (2ab) when squaring a binomial (a + b)² = a² + 2ab + b².
2. Forgetting to Distribute Negative Signs:
Another common mistake occurs when dealing with subtraction. If you have an expression like 3y² - (y² + 14y + 49), you need to distribute the negative sign to every term inside the parentheses. Forgetting to do this will change the signs of the terms incorrectly, leading to the wrong answer. For example, - (y² + 14y + 49) should be -y² - 14y - 49, not -y² + 14y + 49.
3. Combining Non-Like Terms:
This is a classic mistake! Remember, you can only combine terms that have the same variable raised to the same power. For example, you can combine 3y² and y² because they both have y raised to the power of 2. However, you cannot combine 3y² and 14y because one has y² and the other has y. Mixing up like and non-like terms will lead to an incorrect simplification.
4. Not Following the Order of Operations (PEMDAS/BODMAS):
The order of operations is crucial in math. PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) helps us remember the correct sequence of operations. When simplifying polynomials, make sure to handle exponents before multiplication or addition. Ignoring this order can result in incorrect simplifications.
5. Making Arithmetic Errors:
Sometimes, the mistake is as simple as an arithmetic error when adding or subtracting coefficients. For instance, adding 3y² + y² and getting 5y² instead of 4y². These small errors can throw off the entire solution, so it's important to double-check your calculations.
6. Not Writing the Final Answer in Standard Form:
Even if you simplify the polynomial correctly, not writing it in standard form (descending order of exponents) can sometimes lead to confusion or make it harder to match with the correct option in a multiple-choice question. Always ensure your final answer is in the standard form for clarity and correctness.
By keeping these common mistakes in mind, you can improve your accuracy and confidence when simplifying polynomials. Remember to take your time, double-check your work, and break the problem down into manageable steps. Happy simplifying, folks!
Conclusion: Mastering Polynomial Simplification
Well, we've reached the end of our polynomial simplification journey, and what a journey it's been! We started with the polynomial 3y² + (y + 7)² - 15, and through careful step-by-step simplification, we arrived at the standard form 4y² + 14y + 34. We then matched our result with the given options and identified the correct answer as Option C. Simplifying polynomials is a fundamental skill in algebra, and mastering it opens doors to more advanced mathematical concepts.
Throughout this process, we emphasized the importance of understanding the basics: what polynomials are, what standard form means, and how to combine like terms. We broke down the simplification into manageable steps, from expanding the squared term to combining like terms and writing the final answer in standard form. We also highlighted some common mistakes to avoid, such as incorrectly expanding squared terms, forgetting to distribute negative signs, and combining non-like terms. By being aware of these pitfalls, you can significantly improve your accuracy and confidence in algebra.
Simplifying polynomials isn't just about following a set of rules; it's about developing a logical and systematic approach to problem-solving. It's about paying attention to detail, being meticulous with your calculations, and double-checking your work. These skills are not only valuable in mathematics but also in many other areas of life.
So, what’s the takeaway here? Polynomial simplification, like any mathematical skill, requires practice. The more you practice, the more comfortable and confident you'll become. Don't be discouraged by mistakes; they're part of the learning process. Instead, use them as opportunities to learn and improve. Review your work, identify where you went wrong, and try again.
Remember, math is like a language. The more you use it, the more fluent you become. So, keep practicing, keep exploring, and keep challenging yourselves. Whether you're simplifying polynomials, solving equations, or exploring calculus, the journey of mathematical discovery is always rewarding.
And that’s a wrap, guys! We hope this comprehensive guide has helped you understand how to simplify polynomials effectively. Keep up the great work, and we'll see you in the next math adventure!