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Chapter 4: Problem 33

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

Short Answer

Expert verified
25x^2 - 4

Step by step solution

01

Recognize the formula

Identify that this is a difference of squares problem. The given expression (5x + 2)(5x - 2) fits the form of (a + b)(a - b) = a^2 - b^2.
02

Identify the values of a and b

Assign the values to the variables in the formula. Here, a = 5x and b = 2.
03

Apply the difference of squares formula

Substitute the values identified in Step 2 into the difference of squares formula: (5x)^2 - (2)^2. This results in: (5x)^2 - 2^2.
04

Simplify each term

Calculate the squares of each term. (5x)^2 = 25x^2 and 2^2 = 4.
05

Subtract the results

Combine the results from Step 4: 25x^2 - 4. Therefore, the product of (5x + 2)(5x - 2) is 25x^2 - 4.

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Key Concepts

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

algebraic expressions
An algebraic expression is a combination of variables, numbers, and at least one operation. To break it down:**
  • Variables: symbols representing unknown values (e.g., x, y).
    • Expression Example: 5x + 2 and 5x - 2. Use parentheses to clarify operations values.
Recognizing patterns in algebraic expressions helps simplify and solve problems.
polynomial multiplication
Polynomial is an expression with multiple terms. Multiplication involves using distributive property.
Given: (5x + 2)(5x - 2)
  • This fits **difference of squares** pattern.
    • Convert pattern: (a + b)(a - b) = a虏 - b虏:
  • Identify a and b values:
    • a = 5x
    • b = 2
  • Apply formula: a虏 - b虏.
    • Efficiently solve problems by recognizing patterns.
factoring
Factoring transforms expressions into a product. Vital for solving equations. Reverse process of distribution.
Steps:
  • Recognize structure: a虏 - b虏 = (a + b)(a - b)

    • Example: 25x虏 - 4
    Factor: (5x + 2)(5x - 2).
    Simplify higher-degree polynomials.

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Most popular questions from this chapter

If the value of \(x\) is \(6,\) what is the volume of the cube (in cubic units)?

If it costs \(\left(4 x^{5}+3 x^{4}+2 x^{3}+9 x^{2}-29 x+2\right)\) dollars to fertilize a garden, and fertilizer costs \((x+2)\) dollars per square yard, write an expression, in square yards, for the area of the garden.

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. $$ 301 \times 299 $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ 5 k^{2}\left(k^{3}-3\right)\left(k^{2}-k+4\right) $$

Find each product. $$ (5 x+7)(3 y-8) $$
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