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Chapter 4: Problem 20

Find each product. $$ 4 x(5 x+3) $$

Short Answer

Expert verified
The product is \(20x^2 + 12x\).

Step by step solution

01

Distribute the First Term

Multiply the first term outside the parentheses, which is \(4x\), by each term inside the parentheses. The terms inside the parentheses are \(5x\) and \(3\).
02

Multiply \(4x\) by \(5x\)

Multiply the coefficients: \(4\) and \(5\) to get \(20\), then multiply the variables: \(x \cdot x = x^2\). So, \(4x \cdot 5x = 20x^2\).
03

Multiply \(4x\) by \(3\)

Multiply the coefficient \(4\) by \(3\), keeping the variable \(x\): \(4x \cdot 3 = 12x\).
04

Combine the Products

Add the results from Step 2 and Step 3. The expression becomes: \(20x^2 + 12x\).

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Key Concepts

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

Multiplying Variables
When you multiply terms that contain variables, you need to handle the coefficients and the variables separately. The coefficients are the numbers in front of the variables.
When multiplying the variables themselves, you use the property which increases the exponent. For instance, when you multiply 4x and 5x:
  • Multiply the coefficients: 4 and 5 to get 20.
  • Multiply the variables: x and x to get x-squared. Mathematically, it's expressed as: \(x \cdot x = x^2\).

Therefore, 4x * 5x turns into 20x虏.
Coefficients
Coefficients are the numbers that are multiplied by the variables in an algebraic term. In the exercise, 4 and 5 are coefficients.
When multiplying terms with coefficients and variables, you first multiply the coefficients.
  • In the expression 4x * 5x, you first multiply 4 and 5 which equals 20.
  • Then, you handle the variables as explained previously.

Similarly, when multiplying 4x by 3 (another coefficient), you multiply the coefficients 4 and 3 to get 12 and then attach the variable x, resulting in 12x.
Combining Like Terms
After distributing and multiplying, you often end up with multiple terms in your expression. The last important step is combining like terms. Like terms have exactly the same variable parts.
In our example, after distribution and multiplication, we have the terms 20x虏 and 12x.
  • Terms with x虏 and x are not like terms, so you cannot combine these two.
  • Therefore, the final expression remains 20x虏 + 12x.
Make sure to always simplify the expression by combining like terms when possible.

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Most popular questions from this chapter

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. $$ 101 \times 99 $$

\(\frac{\left(3 p^{-2} q^{3}\right)^{2}\left(5 p^{-1} q^{-4}\right)^{-1}}{\left(p^{2} q^{-2}\right)^{-3}}\)

Find each product. $$ -3 r(r-1)\left(r^{2}+r+1\right) $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (m+6)^{2} $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (3 r-2 s)^{4} $$
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