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Chapter 0: Problem 45

List the terms and coefficients of each term. $$ 12 x^{2} y^{5}-x y^{4}+9.2 x y^{3} $$

Short Answer

Expert verified
The terms are 12x^2y^5, -xy^4, and 9.2xy^3 with coefficients 12, -1, and 9.2 respectively.

Step by step solution

01

Identify the first term

The first term in the expression is 12x^2y^5.
02

Identify the coefficient of the first term

The coefficient of the first term is 12.
03

Identify the second term

The second term in the expression is -xy^4.
04

Identify the coefficient of the second term

The coefficient of the second term is -1.
05

Identify the third term

The third term in the expression is 9.2xy^3.
06

Identify the coefficient of the third term

The coefficient of the third term is 9.2.

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Key Concepts

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

identifying terms
Terms in a polynomial are the individual parts that are added or subtracted. Each term consists of numbers, variables, or a combination of both. In the polynomial expression given, we have three terms:
  • First Term: 12虫虏测鈦�
  • Second Term: -虫测鈦�
  • Third Term: 9.2虫测鲁

To identify terms, look for the pieces of the expression that are separated by the plus or minus signs. Each segment between these signs is counted as a term.
coefficients in algebra
Coefficients are the numerical factors of the terms in an algebraic expression. They are the numbers in front of the variables. In the given expression, the coefficients are:
  • The coefficient of the first term: The number 12 in 12虫虏测鈦�
  • The coefficient of the second term: The number -1 in -虫测鈦� (Note that if no number is visible, it is implied to be 1 or -1 if the term is preceded by a minus sign.)
  • The coefficient of the third term: The number 9.2 in 9.2虫测鲁

Remember, the coefficients are crucial because they quantify the term. Without them, the term lacks magnitude.
polynomial expressions
A polynomial expression is a mathematical phrase involving a sum of powers in one or more variables multiplied by coefficients. In our example, 12 x^{2} y^{5}-x y^{4}+9.2 x y^{3}, the polynomial expression is made up of three terms, each with its own set of variables raised to powers.
  • Variables: The letters like x and y are variables that represent numbers.
  • Exponents: The numbers that the variables are raised to, like the 2 in 虫虏 or the 5 in 测鈦�.
  • Coefficients: The numerical factors that multiply the variables and their exponents.

Understanding how polynomial expressions are structured helps in solving them through addition, subtraction, multiplication, or even factoring.

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Most popular questions from this chapter

We know that \((a-b)^{2}=a^{2}-2 a b+b^{2}\). Derive a special product formula for \((a-b)^{3}\).

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (4 x \sqrt{y}-2 y \sqrt{x})(4 x \sqrt{y}+2 y \sqrt{x}) $$

Simplify each expression. $$ (6 x+5)(6 x-5)-(6 x+5)^{2} $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (\sqrt{x+1}-5)^{2} $$

Use a calculator to approximate the expression to 2 decimal places. $$5000\left(1+\frac{0.06}{12}\right)^{(12)(5)}$$
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