If B=p+5 And A=p^2+10p+1 Find An Expression That Equals 2b+3a In Standard Form is a common problem in algebra that challenges students to manipulate and combine algebraic expressions. Understanding how to find a simplified, standard form of an expression like 2b + 3a, given the relationships B=p+5 and A=p^2+10p+1, is fundamental to mastering algebraic concepts. This article will guide you step-by-step through the process of substituting the given expressions for A and B into 2b + 3a, simplifying the expression, and presenting it in standard form. Whether you're studying for an exam or looking to strengthen your algebra skills, this comprehensive guide will help clarify the process.
Understanding the Given Expressions
Before diving into the algebraic manipulation, it’s important to understand the given relationships:
- B = p + 5
- A = p² + 10p + 1
Your goal is to find an expression that equals 2b + 3a in standard form, using these relationships.
What is Standard Form?
Standard form in algebra typically means writing a polynomial in descending powers of the variable, with the leading coefficient positive, and combining like terms. For linear and quadratic expressions, standard form helps clarify the structure and simplifies further calculations.
Step-by-Step Solution Approach
The process involves substituting the given expressions into 2b + 3a, simplifying, and arranging the result in standard polynomial form.
Step 1: Write Down the Expression to Simplify
The expression we are asked to find is:
\[ 2b + 3a \]
Given:
\[ B = p + 5 \]
\[ A = p^2 + 10p + 1 \]
However, note that the variables are B and A, while our expression involves b and a. To avoid confusion, ensure the variables are consistent.
Important clarification: Since the problem provides B and A, but asks for an expression involving b and a, it’s likely that the lowercase variables b and a are intended to correspond to the uppercase B and A, respectively.
Assuming that:
- b = B = p + 5
- a = A = p^2 + 10p + 1
This is a common notation in algebra problems where uppercase variables are given and lowercase variables are used as placeholders.
Step 2: Substitute the Expressions for b and a
Replace b and a in 2b + 3a with their expressions:
\[ 2b + 3a = 2(p + 5) + 3(p^2 + 10p + 1) \]
Step 3: Expand the Expression
Distribute the coefficients:
\[ 2(p + 5) = 2p + 10 \]
\[ 3(p^2 + 10p + 1) = 3p^2 + 30p + 3 \]
Now, the expression becomes:
\[ 2p + 10 + 3p^2 + 30p + 3 \]
Step 4: Combine Like Terms
Group terms according to powers of p:
\[ 3p^2 + (2p + 30p) + (10 + 3) \]
Simplify:
\[ 3p^2 + 32p + 13 \]
This is the simplified form of the expression.
Expressing in Standard Polynomial Form
The expression:
\[ 3p^2 + 32p + 13 \]
is already in standard polynomial form, with terms arranged in decreasing powers of p.
Final Expression in Standard Form:
\[
\boxed{
2b + 3a = 3p^2 + 32p + 13
}
\]
This expression can be used for further calculations or graphing, depending on the context.
Key Takeaways and Tips for Algebraic Manipulation
Understanding Substitution
- Always verify the correspondence between variables. In this problem, lowercase variables b and a are linked to uppercase B and A.
- Substituting the given expressions correctly is crucial for accurate simplification.
Distributive Property
- Use the distributive property to expand expressions with coefficients outside parentheses.
Combining Like Terms
- Group similar powers of the variable to simplify the expression efficiently.
Writing in Standard Form
- Arrange polynomial terms from highest to lowest power.
- Simplify coefficients where possible for clarity.
Additional Practice Problems
To reinforce your understanding, consider working on similar problems:
- Given C = p - 4 and D = p^2 - 3p + 7, find an expression that equals 4c - 2d in standard form.
- If X = 2p + 3 and Y = p^2 - p + 5, find an expression for 3x + 2y in standard form.
- Given E = p^2 + 2p + 3 and F = 3p + 4, find the simplified expression for 5E - 2F.
Practicing these will help solidify your algebra skills and improve problem-solving efficiency.
Conclusion
In summary, to find an expression that equals 2b + 3a in standard form given the relationships B = p + 5 and A = p^2 + 10p + 1, follow these key steps:
- Recognize the variables and substitute the given expressions.
- Expand the expressions using the distributive property.
- Combine like terms carefully.
- Arrange the resulting polynomial in standard form, with descending powers of p.
The resulting simplified expression:
\[ 3p^2 + 32p + 13 \]
is a concise, standard form that accurately represents 2b + 3a based on the given relationships. Mastering these steps enhances your ability to handle similar algebraic problems confidently and efficiently.
Remember: Practice is essential in algebra. The more problems you solve, the more intuitive these steps will become, making complex expressions much easier to simplify and analyze.
