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Chapter 4: Problem 45

Find each product. $$ q(5 q-1)(5 q+1) $$

Short Answer

Expert verified
25q^3 - q

Step by step solution

01

Understand the formula

Recognize that the expression can be rewritten as the product of a quadratic expression and a linear term: q(5q-1)(5q+1).
02

Apply the difference of squares

Recall the difference of squares formula: (a-b)(a+b) = a^2 - b^2. In this problem, let a = 5q and b = 1. Therefore, (5q-1)(5q+1) = (5q)^2 - 1^2.
03

Simplify the expression

Calculate the squares: (5q)^2 = 25q^2 and 1^2 = 1. Thus, (5q-1)(5q+1) = 25q^2 - 1.
04

Multiply by q

Now multiply the simplified expression by q: q(25q^2 - 1). Distribute the q to get: 25q^3 - q.

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Key Concepts

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

Difference of Squares
In the given exercise, we recognize that the terms \(5q\) and \(1\) fit the form of \(a - b\) and \(a + b\). Thus, we set \(a = 5q\) and \(b = 1\). Applying the difference of squares formula converts the expression \( (5q - 1)(5q + 1) \) into \( (5q)^2 - 1^2 = 25q^2 - 1 \). This step greatly simplifies the problem by reducing it to a single quadratic term minus a constant.
Quadratic Expression
Understanding quadratic expressions is essential because they appear frequently in various mathematical contexts. By recognizing that \(25q^2 - 1\) is a quadratic expression, we can also foresee methods to further manipulate or solve equations involving such expressions. When quadratic expressions are in factored form, like our initial expression \(q(5q - 1)(5q + 1)\), it becomes easier to perform multiplication and other algebraic operations.
Distributive Property
We have the quadratic expression \(25q^2 - 1\) that needs to be multiplied by \(q\). According to the distributive property, we multiply \(q\) by each term inside the parenthesis:
  • \(q \times 25q^2 = 25q^3\)
  • \(q \times (-1) = -q\)
Combining these results, we get \(25q^3 - q\), which is the final answer. Using the distributive property allows us to break down and simplify the multiplication process, ensuring the correct expansion and combination of terms.

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Most popular questions from this chapter

Find each product. $$ -5 r(r+1)^{3} $$

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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. $$ 20 \frac{1}{2} \times 19 \frac{1}{2} $$

Find each product. $$ (6-5 x)(2+x) $$
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