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Chapter 4: Problem 29

Find each product. $$ 3 a^{2}\left(2 a^{2}-4 a b+5 b^{2}\right) $$

Short Answer

Expert verified
The product is \(6a^4 - 12a^3b + 15a^2b^2\).

Step by step solution

01

- Apply the Distributive Property

Distribute the term outside the parentheses, which is \(3a^2\), to each term inside the parentheses: \(2a^2\), \(-4ab\), and \(5b^2\).
02

- Multiply \(3a^2\) by Each Term

Perform the multiplication for each term: - \(3a^2 \cdot 2a^2 = 6a^4\) - \(3a^2 \cdot (-4ab) = -12a^3b\) - \(3a^2 \cdot 5b^2 = 15a^2b^2\)
03

- Combine the Products

Write the final expression by combining the products from Step 2: \[ 6a^4 - 12a^3b + 15a^2b^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Multiplication
Algebraic multiplication involves distributing and multiplying terms in expressions. When you see an algebraic expression such as \[3a^2(2a^2 - 4ab + 5b^2)\], the first step is to use the Distributive Property. This property states that you need to multiply the term outside the parentheses by every term inside the parentheses.

Here, we distribute \(3a^2\) to \(2a^2\), \(-4ab\), and \(5b^2\). This gives us three separate multiplication operations to perform. Each multiplication must be handled carefully to ensure all terms are correctly calculated.
Polynomial Expressions
Polynomial expressions are algebraic expressions containing multiple terms. In our example, \(3a^2(2a^2 - 4ab + 5b^2)\) involves a polynomial within the parentheses.

Polynomials can have constants, variables, and exponents. The expression inside the parentheses, \(2a^2 - 4ab + 5b^2\), is a polynomial of three terms:
  • \(2a^2\), which is a term with a squared variable
  • \(-4ab\), which is a term with two different variables multiplied together
  • \(5b^2\), which is a term with another squared variable
Dealing with polynomial expressions often means performing multiple distributions and carefully accounting for all variables and exponents.
Combining Like Terms
Combining like terms is a fundamental part of simplifying algebraic expressions and finalizing the result after distribution. However, in our example, after distributing and multiplying each term in \(3a^2(2a^2 - 4ab + 5b^2)\), we do not actually need to combine like terms because each resulting term is unique.

The expression simplifies to \[6a^4 - 12a^3b + 15a^2b^2\]. To explain further:
  • \(6a^4\) is from \(3a^2 \times 2a^2\)
  • \(-12a^3b\) is from \(3a^2 \times -4ab\)
  • \(15a^2b^2\) is from \(3a^2 \times 5b^2\)
Since the terms \(6a^4, -12a^3b,\text{ and } 15a^2b^2\) are distinct and share no common factors, they are left as is. If you had any terms that were the same, you would combine their coefficients while keeping the variable part intact.

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Most popular questions from this chapter

Graph each equation by completing the table of values. $$ \begin{aligned} &y=2 x^{2}+2\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ 7(4 m-3)(2 m+1) $$

Find each product. $$ 3 y^{3}(2 y+3)(y-5) $$

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 30 \frac{1}{3} \times 29 \frac{2}{3} $$

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications. Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 101 \times 99 $$
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