Problem 17 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ 10 x^{2}-8 x+9 x-9 x^{2} $$

Step-by-Step Solution

Answer

The simplified polynomial is \(x^2 + x\).

Step 1: Combine Like Terms

Identify the like terms in the polynomial. Here, the like terms are those with the same degree. For the given polynomial, the like terms are:\[10x^2 - 9x^2\]and\[-8x + 9x\]. Combine these like terms:\[10x^2 - 9x^2 = 1x^2\]and\[-8x + 9x = 1x\].

Step 2: Write in Descending Order

Once the like terms have been combined, rearrange the simplified terms in descending order of their exponents. The terms in descending order are:\[x^2 + x\]. This expression is already simplified and ordered correctly.

Key Concepts

Combining Like Terms

When working with polynomials, combining like terms is a crucial step that simplifies the expression. Like terms are parts of the polynomial that have the same variable raised to the same power, meaning they can be grouped together using basic arithmetic. For example, in the polynomial \(10x^2 - 8x + 9x - 9x^2\), terms like \(10x^2\) and \(-9x^2\) are considered like terms because they both contain \(x^2\). Similarly, \(-8x\) and \(9x\) are like terms as they both involve \(x\).
To combine these like terms:

• Add or subtract the coefficients (numbers in front of the variables) of the like terms.
• For \(10x^2 - 9x^2\), subtract the coefficients: \(10 - 9 = 1\).
• For \(-8x + 9x\), perform the arithmetic: \(-8 + 9 = 1\).

This results in a simplified expression: \(1x^2 + 1x\), which can further be written as \(x^2 + x\). Combining like terms clarifies the expression, making it easier to manage and understand.

Descending Order

After simplifying a polynomial by combining like terms, it is often required to present the expression in descending order. Descending order in algebra refers to arranging terms in a polynomial starting with the highest degree to the lowest.
Each term in a polynomial has a degree, which is determined by the exponent on its variable. The term with the highest degree is placed first; the next highest follows and so on. For example, in your simplified polynomial \(x^2 + x\):

• \(x^2\) is the highest degree term since the exponent is 2.
• \(x\), having an exponent of 1, follows \(x^2\).

This formatting builds a clear structure making it easier to analyze and apply operations to polynomials, such as differentiation, integration, or further simplification when solving equations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Polynomials are a specific type of algebraic expression that involves terms composed of variables raised to a non-negative integer power.
Understanding algebraic expressions is essential as they form the foundation for algebra and higher mathematics. Here are key components:

• Variables: Symbols such as \(x\) in the polynomial \(10x^2 - 8x + 9x - 9x^2\), representing numbers we may not know.
• Coefficients: Numbers in front of the variables, like 10 and -9 in \(10x^2\) and \(-9x^2\).
• Operators: Signs like \(+\) and \(-\) indicate the mathematical operations.

Mastering algebraic expressions allows students to simplify and solve equations efficiently, paving the way for more complex algebraic operations. This knowledge isn't just theoretical; it's used in various real-world applications, from engineering to data analysis.