Problem 64
Question
Simplify the algebraic expressions for the following problems. $$ a^{3} b^{3} c^{4}\left(4 a+2 b+3 c+a b+a c+b c^{2}\right) $$
Step-by-Step Solution
Verified Answer
Answer: The simplified form of the expression is \(4 a^4 b^3 c^4 + 2 a^3 b^4 c^4 + 3 a^3 b^3 c^5 + a^4 b^4 c^4 + a^4 b^3 c^5 + a^3 b^4 c^6\).
1Step 1: Distribute the term \(a^3 b^3 c^4\) to each term inside the parenthesis
Multiply the term \(a^3 b^3 c^4\) by each term individually inside the parenthesis:
$$
a^{3} b^{3} c^{4}(4a + 2b + 3c + ab + ac + bc^2)
$$
2Step 2: Perform the multiplications
Perform the multiplication for each term:
$$
4 a^4 b^3 c^4 + 2 a^3 b^4 c^4 + 3 a^3 b^3 c^5 + a^4 b^4 c^4 + a^4 b^3 c^5 + a^3 b^4 c^6
$$
Now, the expression is simplified as there are no like terms left to be combined.
Key Concepts
Polynomial MultiplicationExponentsDistributive PropertySimplification
Polynomial Multiplication
Polynomial multiplication involves expanding expressions by applying algebraic multiplication to terms within an expression. In this exercise, you start with an expression like \(a^{3} b^{3} c^{4} (4a + 2b + 3c + ab + ac + bc^2)\).
The goal is to multiply \(a^{3} b^{3} c^{4}\) by each term inside the parentheses. This requires distributing the outside term across each term inside. This process is similar to distributing groceries into different cabinets — each term inside needs to receive a portion of the outside term.
Breaking it down involves handling each component of the polynomial separately and meticulously ensuring all given variables and their powers are covered in this multiplication.
The goal is to multiply \(a^{3} b^{3} c^{4}\) by each term inside the parentheses. This requires distributing the outside term across each term inside. This process is similar to distributing groceries into different cabinets — each term inside needs to receive a portion of the outside term.
Breaking it down involves handling each component of the polynomial separately and meticulously ensuring all given variables and their powers are covered in this multiplication.
Exponents
Exponents are used to represent repeated multiplication of a number by itself. When multiplying terms with exponents, add the exponents of like bases together. For example, multiplying \(a^3\) by \(a\) gives \(a^{4}\) because \(3 + 1 = 4\).
In our expression, you see terms like \(a^3 \times 4a\).
Here, you add the exponents of \(a\), resulting in \(a^4\). The same goes for each of the variables. Remember, the exponent rules are crucial for simplifying these complex expressions.
Ensure you apply these correctly to simplify the expression, as demonstrated in the solution.
In our expression, you see terms like \(a^3 \times 4a\).
Here, you add the exponents of \(a\), resulting in \(a^4\). The same goes for each of the variables. Remember, the exponent rules are crucial for simplifying these complex expressions.
Ensure you apply these correctly to simplify the expression, as demonstrated in the solution.
Distributive Property
The distributive property is key in expanding expressions. It states that for any numbers or variables \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). This means you multiply the term outside the parentheses, across each term inside, just like with spreading frosting evenly on a cake.
The original expression \(a^{3} b^{3} c^{4} (4a + 2b + 3c + ab + ac + bc^2)\) employs this property.
Using the distributive property means:
Continue this systematically for each term inside the parenthesis to simplify the expression, increasing understanding through consistent practice.
The original expression \(a^{3} b^{3} c^{4} (4a + 2b + 3c + ab + ac + bc^2)\) employs this property.
Using the distributive property means:
- \(a^{3} b^{3} c^{4} \times 4a = 4a^4b^3c^4\)
- \(a^{3} b^{3} c^{4} \times 2b = 2a^3b^4c^4\)
Continue this systematically for each term inside the parenthesis to simplify the expression, increasing understanding through consistent practice.
Simplification
Simplification is about making an algebraic expression easier to understand and work with, by reducing it to its simplest form.
After multiplying the terms, you need to look for like terms — terms that have exactly the same variables raised to the same power. Combining like terms can further simplify the expression. However, in this problem, once the distribution and individual multiplication are complete, each term is unique.
This means there are no further simplifications needed beyond the computation of multiplications.
Ensuring your expression is fully simplified is critical for accuracy in algebra, helping both with understanding and when applying these expressions in future problems.
After multiplying the terms, you need to look for like terms — terms that have exactly the same variables raised to the same power. Combining like terms can further simplify the expression. However, in this problem, once the distribution and individual multiplication are complete, each term is unique.
This means there are no further simplifications needed beyond the computation of multiplications.
Ensuring your expression is fully simplified is critical for accuracy in algebra, helping both with understanding and when applying these expressions in future problems.
Other exercises in this chapter
Problem 64
For the following problems, perform the multiplications and combine any like terms. $$ 2 x^{4}\left(6 x^{3}-5 x^{2}-2 x+3\right) $$
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For the following problems, note how many: $$ (x+1) \text { 's in } 8(x+1) ? $$
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For the following problems, find the products. Expand \((a+b)^{2}\) to prove it is equal to \(a^{2}+2 a b+b^{2}\).
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For the following problems, perform the multiplications and combine any like terms. $$ -5 x^{2}(x+2) $$
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