Evaluating Polynomials With Given Values

Hey guys! Today, we're diving into the world of polynomials and how to evaluate them. We'll be tackling a couple of examples that might seem tricky at first, but don't worry, we'll break them down step by step. Whether you're brushing up on your algebra skills or just curious about math, this guide is for you. So, let's jump right in and make those polynomials less intimidating!

Problem 1: Evaluating $6a^2 - 5ab + b^2 - (3a^2 - 5ab + b^2)$ when $a = -2/3$ and $b = -3$

Okay, the first question we're going to look at involves finding the value of a polynomial expression when we have specific values for the variables. This might sound like a mouthful, but it's really just about substituting the numbers and doing the math. Our polynomial expression is $6a^2 - 5ab + b^2 - (3a^2 - 5ab + b^2)$, and we're told that $a = -\frac{2}{3}$ and $b = -3$. So, what do we do? We plug these values into the expression and simplify.

Step-by-Step Solution

1. Write down the original expression: $6a^2 - 5ab + b^2 - (3a^2 - 5ab + b^2)$
2. Substitute the values of a and b: $6(-\frac{2}{3})^2 - 5(-\frac{2}{3})(-3) + (-3)^2 - [3(-\frac{2}{3})^2 - 5(-\frac{2}{3})(-3) + (-3)^2]$
3. Simplify the terms inside the parentheses: First, let's simplify the squares: $(-\frac{2}{3})^2 = \frac{4}{9}$ and $(-3)^2 = 9$. Now substitute these back into the expression: $6(\frac{4}{9}) - 5(-\frac{2}{3})(-3) + 9 - [3(\frac{4}{9}) - 5(-\frac{2}{3})(-3) + 9]$
4. Perform the multiplications: Now, let's multiply each term:
   • $6(\frac{4}{9}) = \frac{24}{9} = \frac{8}{3}$
   • $-5(-\frac{2}{3})(-3) = -10$
   • $3(\frac{4}{9}) = \frac{12}{9} = \frac{4}{3}$ Substitute these back in: $\frac{8}{3} - 10 + 9 - [\frac{4}{3} - 10 + 9]$
5. Simplify inside the brackets: Combine the numbers inside the brackets: $\frac{4}{3} - 10 + 9 = \frac{4}{3} - 1 = \frac{4}{3} - \frac{3}{3} = \frac{1}{3}$ So our expression becomes: $\frac{8}{3} - 10 + 9 - \frac{1}{3}$
6. Combine like terms: Now, let's simplify the expression outside the brackets: $\frac{8}{3} - 10 + 9 = \frac{8}{3} - 1 = \frac{8}{3} - \frac{3}{3} = \frac{5}{3}$ And finally: $\frac{5}{3} - \frac{1}{3} = \frac{4}{3}$

Final Answer

The value of the polynomial $6a^2 - 5ab + b^2 - (3a^2 - 5ab + b^2)$ when $a = -\frac{2}{3}$ and $b = -3$ is $\frac{4}{3}$.

Key Takeaways

• Substitution is Key: Always start by carefully substituting the given values into the expression. It's like plugging in the right ingredients for a recipe.
• Order of Operations: Remember the order of operations (PEMDAS/BODMAS) to avoid mistakes. Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
• Simplify Step-by-Step: Break the problem down into smaller, manageable steps. This makes the whole process less daunting and reduces the chance of errors.

Problem 2: Simplifying $-8a^2 - 2ax - x^2 - (-4a^2 - 2ax - x^2)$

Now, let's tackle another type of polynomial problem. This time, we're not given specific values for the variables. Instead, we need to simplify the expression. This involves combining like terms and getting rid of parentheses. The expression we're working with is $-8a^2 - 2ax - x^2 - (-4a^2 - 2ax - x^2)$. It looks a bit intimidating, but trust me, we can handle it.

Step-by-Step Solution

1. Write down the original expression: $-8a^2 - 2ax - x^2 - (-4a^2 - 2ax - x^2)$
2. Distribute the negative sign: The first thing we need to do is get rid of the parentheses. Notice the negative sign in front of the second set of parentheses? We need to distribute this negative sign to each term inside the parentheses. This means we change the sign of each term: $-8a^2 - 2ax - x^2 + 4a^2 + 2ax + x^2$
3. Identify like terms: Now, let's identify the terms that are similar. Like terms are those that have the same variables raised to the same powers. In this expression, we have:
   • $-8a^2$ and $+4a^2$ (terms with $a^2$)
   • $-2ax$ and $+2ax$ (terms with $ax$)
   • $-x^2$ and $+x^2$ (terms with $x^2$)
4. Combine like terms: Now, we combine these like terms by adding or subtracting their coefficients:
   • $-8a^2 + 4a^2 = -4a^2$
   • $-2ax + 2ax = 0$
   • $-x^2 + x^2 = 0$
5. Write the simplified expression: After combining like terms, we have: $-4a^2 + 0 + 0 = -4a^2$

Final Answer

The simplified form of the polynomial $-8a^2 - 2ax - x^2 - (-4a^2 - 2ax - x^2)$ is $-4a^2$.

Key Takeaways

• Distribute Carefully: When you have a negative sign or any number in front of parentheses, make sure to distribute it correctly to each term inside. It’s a common place to make mistakes.
• Identify Like Terms: Being able to spot like terms is crucial for simplifying expressions. Look for terms with the same variables and exponents.
• Combine with Confidence: Once you've identified like terms, combine them by adding or subtracting their coefficients. Remember the rules for adding and subtracting integers!

Conclusion: Polynomials Aren't So Scary!

So there you have it, guys! We've tackled two different types of polynomial problems: evaluating a polynomial with given values and simplifying a polynomial expression. By breaking down the problems step by step, we've shown that these kinds of questions are totally manageable. Remember, the key is to take your time, be careful with the details, and always double-check your work. Keep practicing, and soon you'll be a polynomial pro!

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