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Chapter 5: Problem 120

Construct three like terms of degree 4.

Short Answer

Expert verified
The three like terms of degree 4 are \(3x^4\), \(5x^2y^2\), and \(-7y^4\).

Step by step solution

01

Understanding Like Terms

Like terms are terms that have the same variables raised to the same powers. For this problem, each term must be of degree 4.
02

Choose the Variables

Select variables to be used in the terms. For example, let's use \( x \) and \( y \).
03

Construct the First Term

Construct the first term with a total degree of 4. One example could be \(3x^4\).
04

Construct the Second Term

Construct the second term with a total degree of 4 using different coefficients. For example, \(5x^2y^2\).
05

Construct the Third Term

Construct the third term with a total degree of 4 using another set of variables and coefficients. For instance, \(-7y^4\).

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Key Concepts

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

Polynomial Expressions
A polynomial expression is a mathematical phrase that includes variables, coefficients, and exponents. These expressions can involve multiple terms combined using addition or subtraction. For example, a polynomial expression can look like this:
  • \(3x^3 - 5x^2 + 7x - 8\)
  • \(2y^4 + 3y^3 - y + 10\)
Each term in these examples consists of a coefficient (a constant number), a variable (like \(x\) or \(y\)), and an exponent (a power which the variable is raised to).
Polynomial expressions are very important in algebra because they can represent a variety of real-world situations and form the building blocks for more complex mathematical concepts.
Algebraic Terms
An algebraic term is a single mathematical expression. It can be a number, a variable, or a combination of both. For example:
  • \(3x^2\)
  • \(-5y\)
  • \(7\)
In these examples, \(3x^2\) contains a coefficient (3) and a variable raised to a power (\(x^2\)), \(-5y\) has a coefficient (-5) and a variable (\(y\)), whereas 7 is a constant term with no variable.
When we talk about like terms, we're referring to terms that have the same variable parts with the same exponents, even if their coefficients are different. For example, \(4x^2\) and \(-6x^2\) are like terms, but \(4x^2\) and \(4y^2\) are not.
Degree of a Term
The degree of a term is the sum of the exponents of all the variables in that term. It's a measure of how 'steep' or 'complex' the term is in polynomial terms.
Let's take some examples to understand this concept better:
  • For the term \(3x^4\), the degree is 4 because the exponent of \(x\) is 4.
  • For \(5x^2y^2\), the degree is calculated by adding the exponents of \(x\) and \(y\), which gives us \(2 + 2 = 4\).
  • In \(-7y^4\), the degree is 4 because the exponent of \(y\) is 4.
When constructing terms of a specific degree, like degree 4, you make sure the sum of the exponents of the variables in each term equals 4. This helps in classifying and understanding polynomial expressions more effectively.

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Most popular questions from this chapter

What is the degree of \(\left(5 m^{5}\right)^{2} ?\)

Write equations for two functions \(f\) and \(g\) such that the domain of \(f+g\) is \(\\{x | x \text { is a real number and } x \neq-2 \text { and } x \neq 5\\}\)

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=3 x-2\\\ &g(x)=2 x-8 \end{aligned} $$

Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D6 contains the formula $$ =\mathrm{Al}-0.2 * \mathrm{B} 1+0.3^{*} \mathrm{Cl} $$ What is the value in \(\mathrm{D} 6\) if the value in \(\mathrm{Al}\) is \(10,\) the value in \(\mathrm{B} 1\) is \(-3,\) and the value in \(\mathrm{Cl}\) is \(30 ?\)

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 3^{5} \square 3^{4} $$
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