Your Answer Should Be A Polynomial In Standard Form.${ (-1 + 2p)(3 - 4p) = \square }$
Understanding the Problem
When dealing with algebraic expressions, it's essential to understand the concept of expanding and simplifying them. In this article, we will focus on expanding and simplifying the given expression: { (-1 + 2p)(3 - 4p) = \square \}
What is a Polynomial in Standard Form?
A polynomial in standard form is a mathematical expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers. The standard form of a polynomial is typically written with the terms arranged in descending order of the powers of the variables.
Expanding the Given Expression
To expand the given expression, we need to apply the distributive property, which states that for any real numbers a, b, and c: a(b + c) = ab + ac. We can apply this property to the given expression by multiplying each term in the first parentheses by each term in the second parentheses.
Step 1: Multiply the First Term in the First Parentheses by Each Term in the Second Parentheses
The first term in the first parentheses is -1, and the terms in the second parentheses are 3 and -4p. We can multiply -1 by each of these terms to get:
-1(3) = -3 -1(-4p) = 4p
Step 2: Multiply the Second Term in the First Parentheses by Each Term in the Second Parentheses
The second term in the first parentheses is 2p, and the terms in the second parentheses are 3 and -4p. We can multiply 2p by each of these terms to get:
2p(3) = 6p 2p(-4p) = -8p^2
Step 3: Combine Like Terms
Now that we have multiplied each term in the first parentheses by each term in the second parentheses, we can combine like terms to simplify the expression. The like terms are the terms that have the same variable raised to the same power.
-3 + 4p + 6p - 8p^2
Step 4: Simplify the Expression
We can simplify the expression by combining the like terms:
-3 + 10p - 8p^2
The Final Answer
The final answer is a polynomial in standard form, which is:
-8p^2 + 10p - 3
Conclusion
In this article, we have expanded and simplified the given expression using the distributive property and combining like terms. We have also discussed the concept of a polynomial in standard form and how it is typically written with the terms arranged in descending order of the powers of the variables. By following these steps, we can expand and simplify algebraic expressions with ease.
Tips and Tricks
- When expanding and simplifying algebraic expressions, it's essential to apply the distributive property and combine like terms.
- Make sure to arrange the terms in descending order of the powers of the variables to write the polynomial in standard form.
- Use the distributive property to multiply each term in the first parentheses by each term in the second parentheses.
- Combine like terms to simplify the expression.
Real-World Applications
Expanding and simplifying algebraic expressions has numerous real-world applications in various fields, including:
- Science: Algebraic expressions are used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
- Engineering: Algebraic expressions are used to design and optimize systems, such as bridges and buildings.
- Economics: Algebraic expressions are used to model economic systems and make predictions about future trends.
Common Mistakes
- Not applying the distributive property: Failing to apply the distributive property can lead to incorrect results.
- Not combining like terms: Failing to combine like terms can lead to incorrect results.
- Not arranging terms in descending order: Failing to arrange terms in descending order of the powers of the variables can lead to incorrect results.
Conclusion
In conclusion, expanding and simplifying algebraic expressions is a crucial skill in mathematics and has numerous real-world applications. By following the steps outlined in this article, we can expand and simplify algebraic expressions with ease. Remember to apply the distributive property, combine like terms, and arrange terms in descending order of the powers of the variables to write the polynomial in standard form.
Q: What is the distributive property, and how is it used in expanding algebraic expressions?
A: The distributive property is a mathematical concept that states that for any real numbers a, b, and c: a(b + c) = ab + ac. It is used in expanding algebraic expressions by multiplying each term in the first parentheses by each term in the second parentheses.
Q: How do I apply the distributive property to expand an algebraic expression?
A: To apply the distributive property, multiply each term in the first parentheses by each term in the second parentheses. For example, if we have the expression (a + b)(c + d), we would multiply a by c, a by d, b by c, and b by d.
Q: What is the difference between expanding and simplifying an algebraic expression?
A: Expanding an algebraic expression involves applying the distributive property to multiply each term in the first parentheses by each term in the second parentheses. Simplifying an algebraic expression involves combining like terms to reduce the expression to its simplest form.
Q: How do I combine like terms in an algebraic expression?
A: To combine like terms, identify the terms that have the same variable raised to the same power. Then, add or subtract the coefficients of these terms to simplify the expression.
Q: What is a polynomial in standard form, and how is it written?
A: A polynomial in standard form is a mathematical expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers. It is typically written with the terms arranged in descending order of the powers of the variables.
Q: How do I write a polynomial in standard form?
A: To write a polynomial in standard form, arrange the terms in descending order of the powers of the variables. For example, if we have the expression 3x^2 + 2x + 1, we would write it as 3x^2 + 2x + 1.
Q: What are some common mistakes to avoid when expanding and simplifying algebraic expressions?
A: Some common mistakes to avoid include:
- Not applying the distributive property
- Not combining like terms
- Not arranging terms in descending order of the powers of the variables
Q: How do I check my work when expanding and simplifying algebraic expressions?
A: To check your work, plug in a value for the variable and evaluate the expression. If the result is correct, then your work is correct.
Q: What are some real-world applications of expanding and simplifying algebraic expressions?
A: Expanding and simplifying algebraic expressions has numerous real-world applications in various fields, including:
- Science: Algebraic expressions are used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
- Engineering: Algebraic expressions are used to design and optimize systems, such as bridges and buildings.
- Economics: Algebraic expressions are used to model economic systems and make predictions about future trends.
Q: How can I practice expanding and simplifying algebraic expressions?
A: You can practice expanding and simplifying algebraic expressions by working through exercises and problems in a textbook or online resource. You can also try creating your own problems and solving them to practice your skills.
Q: What are some online resources for learning about expanding and simplifying algebraic expressions?
A: Some online resources for learning about expanding and simplifying algebraic expressions include:
- Khan Academy: Khan Academy has a comprehensive video series on algebraic expressions, including expanding and simplifying.
- Mathway: Mathway is an online math problem solver that can help you practice expanding and simplifying algebraic expressions.
- Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you practice expanding and simplifying algebraic expressions.
Q: How can I get help if I'm struggling with expanding and simplifying algebraic expressions?
A: If you're struggling with expanding and simplifying algebraic expressions, you can try:
- Asking a teacher or tutor for help
- Working with a study group or classmate
- Using online resources, such as video tutorials or online calculators
- Practicing regularly to build your skills and confidence.