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Chapter 4: Problem 18

Find each product. $$ 2 x(7 x-2)^{2} $$

Short Answer

Expert verified
The product is \text{\text\( 98x^{3} - 56x^{2} + 8x \)}.

Step by step solution

01

- Expand the Squared Term

First, expand the squared term \text{\text\( (7x - 2)^{2} \)}. Use the binomial expansion formula: \text{\((a - b)^{2} = a^{2} - 2ab + b^{2}\)}. Here, \text{\text\( 7x \)} is \text{\text\(a\)} and \text{\text\( 2 \)} is \text{\text\(b\)}. So, \text{\text\( (7x - 2)^{2} = (7x)^{2} - 2 \times 7x \times 2 + 2^{2} \)}.
02

- Simplify the Binomial Expansion

Simplify the expression from the binomial expansion: \text{\text\( (7x)^{2} = 49x^{2} \)}, \text{\text\( 2 \times 7x \times 2 = 28x \)}, and \text{\text\( 2^{2} = 4 \)}. So, \text{\text\( (7x - 2)^{2} = 49x^{2} - 28x + 4 \)}.
03

- Multiply by 2x

Now multiply the expanded form by \text{\text\( 2x \)}. Distribute \text{\text\(2x\)} to each term inside the parenthesis: \text{\text\( 2x(49x^{2} - 28x + 4) \)}.
04

- Perform the Multiplication

Multiply each term: \text{\text\( 2x \times 49x^{2} = 98x^{3} \)}, \text{\text\( 2x \times -28x = -56x^{2} \)}, and \text{\text\( 2x \times 4 = 8x \)}. The resulting expression is \text{\text\( 98x^{3} - 56x^{2} + 8x \)}.

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Key Concepts

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

Binomial Expansion
Binomial expansion is a method used to expand expressions that are squared or raised to any power. In this exercise, the term \( (7x - 2)^2 \) needs expansion. The formula to remember is \((a - b)^2 = a^2 - 2ab + b^2\). This formula helps break down the expression into simpler terms. When you identify \(a = 7x\) and \(b = 2\), the expansion provides the terms \((7x)^2 = 49x^2\),\( -2(7x)(2) = -28x\), and \(2^2 = 4\). Adding these up gives \(49x^2 - 28x + 4\). It's like unpacking a box, bit by bit.
Distributive Property
The distributive property is fundamental in algebra. It helps distribute a single term across a binomial or polynomial. In this case, we use it to handle the term \(2x\) multiplied by \(49x^2 - 28x + 4\). This property states: \(a(b+c) = ab + ac\). Applied here, \(2x\) gets distributed to each term in the polynomial. So, you do:
  • \(2x \times 49x^2 = 98x^3\)
  • \(2x \times -28x = -56x^2\)
  • \(2x \times 4 = 8x\)
This step simplifies and combines all partial products into one polynomial.
Polynomial Simplification
Simplifying polynomials is the final part of this exercise. It's about combining like terms to create the simplest form. Here, after multiplying, we get \(98x^3 - 56x^2 + 8x\). Notice there are no like terms left, which means each term is unique to its degree (the exponent of \(x\)). Always check if terms can be combined. Typically, like terms have the same variable raised to the same power. In our expression, the terms \(98x^3\), \(-56x^2\), and \(8x\) are already simplified as much as possible. Always end by double-checking the polynomial for any further simplification.

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