Write the monomial in its standard form. Name its coefficient and identify its degree:
[tex]$2 \frac{1}{3} a^2 x\left(-\frac{3}{7}\right) a^3 x^2$[/tex]
Answer (2)
Convert the mixed number to an improper fraction: 2 R A C 1 3 = R A C 7 3 .
Multiply the coefficients: R A C 7 3 \tIMES ( − R A C 3 7 ) = − 1 .
Multiply the variables with the same base by adding their exponents: a 2 \tIMES a 3 = a 5 and x \tIMES x 2 = x 3 .
Write the monomial in standard form, identify the coefficient, and state the degree: − a 5 x 3 , coefficient is -1, degree is 8. − a 5 x 3 , − 1 , 8
Explanation
Understanding the Problem We are given the monomial 2 R A C 1 3 a 2 x ( − R A C 3 7 ) a 3 x 2 and we want to write it in standard form, name its coefficient, and identify its degree.
Converting to Improper Fraction First, convert the mixed number to an improper fraction: 2 R A C 1 3 = R A C 7 3 . So the monomial becomes R A C 7 3 a 2 x ( − R A C 3 7 ) a 3 x 2 .
Multiplying Coefficients Next, multiply the coefficients: R A C 7 3 \tIMES ( − R A C 3 7 ) = − 1 .
Multiplying Variables Now, multiply the variables with the same base by adding their exponents: a 2 \tIMES a 3 = a 2 + 3 = a 5 and x \tIMES x 2 = x 1 + 2 = x 3 .
Writing in Standard Form Write the monomial in standard form: − 1 a 5 x 3 = − a 5 x 3 .
Identifying the Coefficient Identify the coefficient: The coefficient is -1.
Identifying the Degree Identify the degree of the monomial by adding the exponents of the variables: 5 + 3 = 8 .
Final Answer Therefore, the monomial in standard form is − a 5 x 3 , its coefficient is -1, and its degree is 8.
Examples
Monomials are used in various fields, such as physics and engineering, to describe physical quantities. For example, the kinetic energy of an object can be expressed as a monomial involving mass and velocity. Understanding how to simplify and analyze monomials is crucial for solving problems in these fields. In computer graphics, monomials can represent polynomial surfaces and curves, which are fundamental in creating realistic images and animations. By manipulating monomials, designers can control the shape and appearance of objects in virtual environments.
The monomial 2 3 1 a 2 x ( − 7 3 ) a 3 x 2 simplifies to − a 5 x 3 , with a coefficient of -1 and a degree of 8.
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