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Chapter 8: Problem 2

Fill in the blanks. When we factor a polynomial, we write a sum of terms as a ____ of factors.

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01

Understanding Factoring in Mathematics

Factoring is a process of breaking down an expression into products of simpler expressions. In simple terms, it involves rewriting a polynomial as a product of its factors. Factors are expressions that, when multiplied together, give the original expression.
02

Identifying Key Mathematical Terms

In the given sentence, 'sum of terms' refers to the original polynomial which is composed of several terms added together. The word we need to fill in relates to the resulting expression after factoring the polynomial.
03

Completing the Sentence with the Correct Term

After understanding that factoring involves rewriting a polynomial as a product of factors, we can complete the blank with the word 'product.' Thus, the complete sentence is: 'When we factor a polynomial, we write a sum of terms as a product of factors.'

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Key Concepts

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

Polynomial Expression
In mathematics, a polynomial expression is a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient. These expressions can have one or more variables. Here are key characteristics of polynomial expressions:
  • **Coefficients**: The numbers that multiply the variables.
  • **Variables**: Symbols like \(x\), \(y\), etc., that can hold different values.
  • **Exponents**: Non-negative integers showing the power to which a variable is raised.
For example, the expression \(3x^2 + 2x - 5\) is a polynomial. It contains three terms: \(3x^2\), \(2x\), and \(-5\). Factoring such an expression involves rewriting it as a product of simpler polynomials or numbers.
Product of Factors
When you factor a polynomial, you transform it from a sum of terms into a product of its factors. This is essential in simplifying expressions and solving polynomial equations. Let's break this down further:
  • **Factors**: These are expressions or numbers that, when multiplied, give the original polynomial.
  • **Product**: The result of multiplying two or more factors together.
Consider the polynomial \(x^2 - 5x + 6\). Factoring involves writing it as \((x - 2)(x - 3)\). Thus, the factors of the polynomial are \(x - 2\) and \(x - 3\), and the product of these factors equals the original polynomial expression. Factoring simplifies solving equations because setting the product equal to zero allows us to solve for the variable quickly.
Mathematical Terms
In understanding polynomials and their factors, various mathematical terms come into play. Familiarizing yourself with these is crucial for mastering the topic:
  • **Term**: Each part of a polynomial separated by a plus or minus sign.
  • **Constant**: A term with no variable, essentially just a number.
  • **Degree**: The highest exponent of the variable in the polynomial. Influences the polynomial's behavior and complexity.
For instance, in the polynomial \(4x^3 + x^2 - 3\), the term \(4x^3\) has a degree of 3, making it the leading term. Understanding terms, constants, and degrees helps in dissecting and factoring polynomials effectively.

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Most popular questions from this chapter

Environmental Cleanup. Suppose the cost (in dollars) of removing \(p \%\) of the pollution in a river is given by the rational function $$f(p)=\frac{50,000 p}{100-p} \text { where } 0 \leq p<100$$ Find the cost of removing each percent of pollution. a. \(50 \%\) b. \(80 \%\)

Perform the operations and simplify, if possible. See Example \(8 .\) $$\frac{x+5}{x y}-\frac{x-1}{x^{2} y}$$

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{2}{x-1}-\frac{2 x}{x^{2}-1}-\frac{x}{x^{2}+2 x+1}$$

Find two rational expressions, each with denominator \(x^{2}+5 x+6,\) such that their sum is \(\frac{1}{x+2}\)

Perform the operations and simplify, if possible. See Example 9. $$\frac{6}{5 d^{2}-5 d}-\frac{3}{5 d-5}$$
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