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Chapter 9: Problem 43

Expand and multiply. $$(2 x+3)^{2}$$

Short Answer

Expert verified
The expanded form is \(4x^2 + 12x + 9\).

Step by step solution

01

Write the expression as a product

The expression \( (2x+3)^2 \) can be rewritten as \( (2x+3)(2x+3) \). This represents the binomial multiplied by itself.
02

Apply the distributive property

Use the distributive property to multiply each term in the first binomial by each term in the second binomial: \( (2x+3)(2x+3) = 2x(2x) + 2x(3) + 3(2x) + 3(3) \).
03

Simplify each multiplication

Perform each multiplication separately: \( 2x imes 2x = 4x^2 \), \( 2x imes 3 = 6x \), \( 3 imes 2x = 6x \), \( 3 imes 3 = 9 \).
04

Combine like terms

Combine the like terms that result from the multiplication: \( 4x^2 + 6x + 6x + 9 = 4x^2 + 12x + 9 \).

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra. It is used to simplify expressions and equations by distributing a single term across terms inside a parenthesis. For example, if you have an expression like \(a(b + c)\), the distributive property lets you rewrite it as \(ab + ac\).

When applied to polynomials, the distributive property ensures that every term in one polynomial multiplies with every term in the other polynomial. In our given problem, we use the distributive property to rewrite \((2x+3)^2\) as \((2x+3)(2x+3)\). This sets the stage for us to multiply two binomials seamlessly.
Multiplying Polynomials
Multiplying polynomials involves a systematic process of multiplication and combining terms. It's similar to using the distributive property, but we parse through larger expressions.

With binomials like \((2x+3)(2x+3)\), we apply the distributive property to multiply each term in the first binomial with each term in the second. Here's a breakdown:
  • Multiply \(2x\) by each term in the second binomial: \(2x(2x) + 2x(3)\).
  • Multiply \(3\) by each term in the second binomial: \(3(2x) + 3(3)\).
The result is a series of products: \(4x^2 + 6x + 6x + 9\).

This demonstrates how polynomial multiplication expresses the product as a sum of simpler, manageable terms.
Combining Like Terms
Once you've multiplied polynomials, you'll likely end up with terms that can be combined, known as like terms. Like terms are terms that have the same variable raised to the same power. They can be added or subtracted from each other.

In our example of \(4x^2 + 6x + 6x + 9\), the like terms \(6x\) and \(6x\) can be combined. Add these terms together: \(6x + 6x = 12x\).

After combining, we arrive at a simplified polynomial: \(4x^2 + 12x + 9\).

This simplification step is crucial as it provides the cleanest form of your answer, making complex expressions manageable and easier to interpret.

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Most popular questions from this chapter

Multiply the following expressions. $$(2 x-5)(3 x-4)$$

Multiply the following expressions. $$8 x^{2} y^{2}\left(5 x^{3}+4 y^{3}\right)$$

Complete the following table. $$\begin{array}{|c|}\hline \text { Number } & \text { Square } \\\\\hline x & x^2 \\\\\hline \frac{1}{2} & \\\\\hline \frac{1}{3} & \\\\\hline \frac{1}{4} \\\\\hline \frac{1}{5} \\\\\hline \frac{1}{6} \\\\\hline \frac{1}{7} \\\\\hline\end{array}$$

Expand and multiply. $$(2 x-4)^{2}$$

Simplify each expression, and write all answers in scientific notation. $$\frac{(0.00087)(40,000)}{1,160,000}$$
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