Consider The Expression { -3(y-5)^2-9+7y$}$.Which Statements Are True About The Process And Simplified Product? Check All That Apply.- The First Step In Simplifying Is To Distribute The -3 Throughout The Parentheses.- There Are 3 Terms In The
Understanding the Expression
The given expression is {-3(y-5)^2-9+7y$}$. This expression involves a quadratic term, a constant term, and a linear term. To simplify this expression, we need to apply the rules of algebraic manipulation.
Statement 1: Distributing the Coefficient
The first step in simplifying the expression is to distribute the coefficient -3 throughout the parentheses. This means multiplying the coefficient -3 by each term inside the parentheses.
{-3(y-5)^2 = -3(y^2 - 10y + 25)$]
Using the distributive property, we can expand the expression as follows:
[$-3(y^2 - 10y + 25) = -3y^2 + 30y - 75$]
Now, let's rewrite the original expression with the distributed coefficient:
[$-3(y-5)^2-9+7y = -3y^2 + 30y - 75 - 9 + 7y$]
Statement 2: Combining Like Terms
The next step is to combine like terms. In this expression, we have two linear terms: 30y and 7y. We can combine these terms by adding their coefficients.
[$30y + 7y = 37y$]
Now, let's rewrite the expression with the combined like terms:
[$-3y^2 + 37y - 75 - 9$]
Statement 3: Simplifying the Constant Terms
The final step is to simplify the constant terms. In this expression, we have two constant terms: -75 and -9. We can combine these terms by adding their coefficients.
[$-75 - 9 = -84$]
Now, let's rewrite the expression with the simplified constant terms:
[$-3y^2 + 37y - 84$]
Conclusion
In conclusion, the first step in simplifying the expression is to distribute the coefficient -3 throughout the parentheses. The resulting expression is [-3y^2 + 37y - 84$]. Therefore, the correct statements are:
- The first step in simplifying is to distribute the -3 throughout the parentheses.
- There are 3 terms in the simplified product.
Key Takeaways
- Distributing the coefficient throughout the parentheses is the first step in simplifying the expression.
- Combining like terms is an essential step in simplifying the expression.
- Simplifying the constant terms is the final step in simplifying the expression.
Common Mistakes to Avoid
- Failing to distribute the coefficient throughout the parentheses.
- Failing to combine like terms.
- Failing to simplify the constant terms.
Real-World Applications
- Simplifying algebraic expressions is a crucial skill in mathematics and science.
- Understanding how to simplify expressions can help you solve complex problems in fields like physics, engineering, and computer science.
Practice Problems
- Simplify the expression [2(x+3)^2-4+5x\$}.
- Simplify the expression {-4(y-2)^2+3+2y$}$.
Conclusion
Q: What is the first step in simplifying an algebraic expression?
A: The first step in simplifying an algebraic expression is to distribute the coefficient throughout the parentheses. This means multiplying the coefficient by each term inside the parentheses.
Q: How do I distribute the coefficient throughout the parentheses?
A: To distribute the coefficient, you need to multiply the coefficient by each term inside the parentheses. For example, if the expression is {-3(y-5)^2$}$, you would multiply -3 by each term inside the parentheses: {-3y^2 + 30y - 75$}$.
Q: What is the next step in simplifying an algebraic expression?
A: The next step in simplifying an algebraic expression is to combine like terms. This means adding or subtracting terms that have the same variable and exponent.
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients of the terms that have the same variable and exponent. For example, if the expression is $2x + 3x\$}, you would combine the terms by adding their coefficients$.
Q: What is the final step in simplifying an algebraic expression?
A: The final step in simplifying an algebraic expression is to simplify the constant terms. This means adding or subtracting the constant terms to get a single constant term.
Q: How do I simplify the constant terms?
A: To simplify the constant terms, you need to add or subtract the constant terms. For example, if the expression is {-3 + 2$}$, you would simplify the constant terms by adding them: {-1$}$.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Failing to distribute the coefficient throughout the parentheses.
- Failing to combine like terms.
- Failing to simplify the constant terms.
Q: How can I practice simplifying algebraic expressions?
A: You can practice simplifying algebraic expressions by working through practice problems. You can find practice problems in algebra textbooks or online resources.
Q: What are some real-world applications of simplifying algebraic expressions?
A: Simplifying algebraic expressions has many real-world applications, including:
- Solving complex problems in physics and engineering.
- Modeling population growth and decay.
- Analyzing data and making predictions.
Q: Can you provide some examples of simplifying algebraic expressions?
A: Here are some examples of simplifying algebraic expressions:
- Simplify the expression ${2(x+3)^2-4+5x\$}.
- Simplify the expression {-4(y-2)^2+3+2y$}$.
Q: How can I use technology to simplify algebraic expressions?
A: You can use technology, such as calculators or computer software, to simplify algebraic expressions. Many calculators and computer programs have built-in functions for simplifying expressions.
Q: What are some tips for simplifying algebraic expressions?
A: Here are some tips for simplifying algebraic expressions:
- Read the expression carefully and identify the terms.
- Distribute the coefficient throughout the parentheses.
- Combine like terms.
- Simplify the constant terms.
- Check your work for errors.
Conclusion
In conclusion, simplifying algebraic expressions is a crucial skill in mathematics and science. By following the steps outlined in this article, you can simplify expressions and solve complex problems. Remember to distribute the coefficient throughout the parentheses, combine like terms, and simplify the constant terms. With practice, you can become proficient in simplifying algebraic expressions and tackle complex problems with confidence.