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Chapter 5: Problem 64

Mixed Practice Multiply. $$ (x-10)(x+10) $$

Short Answer

Expert verified
The expanded form is \(x^2 - 100\).

Step by step solution

01

Recognize the Pattern

The expression \((x - 10)(x + 10)\) follows the special product pattern known as the difference of squares.
02

Apply the Difference of Squares Formula

The difference of squares formula is \((a - b)(a + b) = a^2 - b^2\). In your case, identify \(a = x\) and \(b = 10\).
03

Substitute into the Formula

Substituting into the formula gives \(x^2 - 10^2\).
04

Simplify the Expression

Calculate \(10^2\), which is 100, so the expression becomes \(x^2 - 100\).

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Key Concepts

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

Difference of Squares
In algebra, the difference of squares is a helpful tool for simplifying certain expressions. It's recognized when you have two terms, both being perfect squares, separated by a subtraction sign. The structure
  • One squared term minus another squared term
  • Expressed as \(a^2 - b^2\)
This pattern allows us to quickly factor the expression into a product of two binomials. When you see something like
  • \((x - 10)(x + 10)\)
it fits perfectly into this category because of its resemblance to
  • \((a - b)(a + b)\)
which equals
  • \(a^2 - b^2\)
Applying this understanding simplifies expressions greatly.
Special Product Patterns
Special product patterns are shortcuts in algebra that help in multiplying expressions more efficiently. They include formulas like
  • the difference of squares
  • the square of a binomial
  • the sum and difference of cubes
In the given example
  • \((x - 10)(x + 10)\)
it demonstrates the special product pattern of the difference of squares. Using this pattern, you can directly obtain \(x^2 - 100\) from
  • \((x - 10)(x + 10)\)
without expanding the terms individually. This saves both time and effort and reduces the chances of errors in calculations.
Factoring Expressions
Factoring is breaking down an expression into a product of simpler components. In algebra, it鈥檚 a necessary skill for simplifying expressions and solving equations. Here鈥檚 how you do it:
  • Identify a pattern鈥攊n this case, a difference of squares.
  • Substitute the terms into the pattern; \((x - 10)(x + 10)\) becomes \(x^2 - 10^2\).
  • Simplify by calculating, \(10^2 = 100\), resulting in the expression \(x^2 - 100\).
Factoring is crucial for mathematical problem-solving and aids in understanding the relationships between numbers and expressions. Once you learn to spot these patterns, working with polynomials becomes much more straightforward.

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Most popular questions from this chapter

Subtract using a vertical format. $$ \begin{array}{r} 5 x^{3}-4 x^{2}+6 x-2 \\ -\left(3 x^{3}-2 x^{2}-x-4\right) \\ \hline \end{array} $$

Subtract using a vertical format. $$ \begin{array}{r} 7 a^{2}-9 a+6 \\ -\left(11 a^{2}-4 a+2\right) \\ \hline \end{array} $$

Evaluate each expression using exponential rules. Write each result in standard form. $$ \left(2.5 \times 10^{6}\right)\left(2 \times 10^{-6}\right) $$

Subtract. $$ \left(-7 y^{2}+5\right)-\left(-8 y^{2}+12\right) $$

The polynomial \(-0.92 x^{2}+2.43 x+34.85\) represents the number of Americans (in millions) under age 65 covered by public health programs during \(1999-2007\). The polynomial \(0.07 x^{2}-0.64 x+180.96\) represents the number of Americans (in millions) under age 65 covered by private health insurance during 1999-2007. In both polynomials, \(x\) represents the number of years since \(1999 .\) Find a polynomial for the total number of Americans (in millions) under age 65 with some form of health coverage during this period. (Source: Based on data from U.S. Census Bureau)
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