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Chapter 5: Problem 27

Combine like terms. $$ 3 x^{2} y-2 x y^{2}+x^{2}+5 x $$

Short Answer

Expert verified
No like terms can be combined, so the solution is \(3x^2y - 2xy^2 + x^2 + 5x\).

Step by step solution

01

Identify like terms

Like terms are terms that have the same variables raised to the same power. In the expression \(3x^2y - 2xy^2 + x^2 + 5x\), the like terms are: \(3x^2y\), \(-2xy^2\), \(x^2\), and \(5x\).
02

Group like terms

Rewrite the expression by grouping the like terms together: \( (3x^2y) + (-2xy^2) + (x^2) + (5x) \).
03

Simplify each group of like terms

Since no like terms can be combined further, the expression remains the same: \(3x^2y - 2xy^2 + x^2 + 5x\).

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Key Concepts

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

like terms
In algebra, 'like terms' are terms that have identical variables raised to the same powers. For instance, in the expression \(3x^2y - 2xy^2 + x^2 + 5x\), each term has a variable part: \(3x^2y\), \(-2xy^2\), \(x^2\), and \(5x\). Like terms share the same variable components. For example:
  • \(x^2\) and \(3x^2y\) have the same power of \(x\), though they are not like terms because they have different variables overall.
  • \(x\) and \(5x\) both have the same variable \(x\), and the same power of 1, making them like terms but there鈥檚 no counterpart in this specific example.
Identifying and combining like terms makes expressions simpler and easier to work with.
variables
Variables are symbols, typically letters like \(x\) or \(y\), that represent numbers in algebraic expressions. They are placeholders for numbers that can change or that are not yet known. In the given expression \(3x^2y - 2xy^2 + x^2 + 5x\):
  • \(x\) and \(y\) are the variables.
  • Each term in the expression has one or more variables associated with it.
Understanding variables helps us to construct, interpret, and manipulate algebraic expressions. Operations on variables follow specific algebraic rules that preserve equality and simplify problems.
polynomials
A polynomial is an algebraic expression composed of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents on variables. For example, in the expression \(3x^2y - 2xy^2 + x^2 + 5x\), we see a combination of terms:
  • \(3x^2y\) and \(-2xy^2\) are polynomial terms with two variables each.
  • \(x^2\) and \(5x\) are univariate polynomial terms (involving only one variable).
Polynomials can be classified by the number of terms they have, and by the degree, which is the highest power of the variable. In this example, a three-term polynomial like \(3x^2y\) forms part of a more complex multi-term polynomial.
algebraic expressions
An algebraic expression is a combination of numbers, variables, and operations. Expressions represent a quantity and can be simplified by performing algebraic operations like addition and multiplication. For example, the expression \(3x^2y - 2xy^2 + x^2 + 5x\) consists of:
  • Terms: \(3x^2y, -2xy^2, x^2, 5x\).
  • Operations: Subtraction, addition.
Algebraic expressions can be simplified by combining like terms and using algebraic rules. Simplifying makes expressions easier to understand and solve, especially when we set them equal to an equation.

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Most popular questions from this chapter

Match the description with the most appropriate algebraic expression from the column on the right. ____A polynomial with degree 5. a) \(8 x^{3}+\frac{2}{x^{2}}\) b) \(5 x^{4}+3 x^{3}-4 x+7\) c) \(\frac{3}{x}-6 x^{2}+9\) d) \(8 t-4 t^{5}\) e) 5 f) \(6 x^{2}+7 x^{4}-2 x^{3}\) g) \(4 t-2 t^{7}\) h) \(3 t^{2}+4 t+7\)

Identify the base and the exponent in each expression. $$ -6 n^{0} $$

Add $$ \begin{aligned} &0.15 x^{4}+0.10 x^{3}-0.9 x^{2}\\\ &\begin{array}{r} {-0.01 x^{3}+0.01 x^{2}+x} \\ {1.25 x^{4} \quad+0.11 x^{2} \quad+0.01} \\ {0.27 x^{3} \quad+0.99} \\ \hline \end{array} \end{aligned} $$

Simplify. Write \(81^{3} \cdot 27 \div 9^{2}\) as a power of 3

Add or subtract, as indicated. $$ \left(5 a^{2} b+7 a b\right)+\left(9 a^{2} b-5 a b\right)+\left(a^{2} b-6 a b\right) $$
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