Problem 40
Question
For the following problems, simplify each of the algebraic expressions. $$ -2 z^{3}+15 z+4 z^{3}+z^{2}-6 z^{2}+z $$
Step-by-Step Solution
Verified Answer
Answer: The simplified form of the given algebraic expression is \(2z^3 - 5z^2 + 16z\).
1Step 1: 1. Identify like terms
In the given expression, we see terms that have the same variables and exponents. Group all similar terms together.
$$
(-2 z^3 + 4 z^3) + (z^2 - 6 z^2) + (15 z + z)
$$
2Step 2: 2. Combine like terms
Now, perform the necessary additions and subtractions for each group,
$$
(-2 z^3 + 4 z^3) = 2 z^3
$$
$$
(z^2 - 6 z^2) = -5 z^2
$$
and
$$
(15 z + z) = 16 z
$$
3Step 3: 3. Write the simplified expression
Combine the simplified groups of terms to create the final expression:
$$
2 z^3 - 5 z^2 + 16 z
$$
The simplified version of the original expression is \(2z^3 - 5z^2 + 16z\).
Key Concepts
Combining Like TermsAlgebraic ManipulationElementary Algebra
Combining Like Terms
One of the fundamental skills in algebra is combining like terms. Like terms are terms that have identical variable parts with the same exponents. For instance, in the expression \( -2 z^{3}+4 z^{3} \) both terms have the variable part \( z^{3} \), making them like terms.
When simplifying expressions, you can think of like terms as items that are identical and therefore can be combined. It's similar to adding apples to apples; you wouldn't add apples to oranges because they are not the same. In algebra, you combine like terms by adding or subtracting their coefficients (the numerical part of the term).
In the given exercise, like terms are grouped and combined, e.g., \( -2 z^{3}+4 z^{3} \) simplifies to \(2 z^{3}\), because \( -2+4 = 2 \). It’s important to pay attention to signs (positive and negative), as they will determine whether you are adding or subtracting the coefficients.
When simplifying expressions, you can think of like terms as items that are identical and therefore can be combined. It's similar to adding apples to apples; you wouldn't add apples to oranges because they are not the same. In algebra, you combine like terms by adding or subtracting their coefficients (the numerical part of the term).
In the given exercise, like terms are grouped and combined, e.g., \( -2 z^{3}+4 z^{3} \) simplifies to \(2 z^{3}\), because \( -2+4 = 2 \). It’s important to pay attention to signs (positive and negative), as they will determine whether you are adding or subtracting the coefficients.
Algebraic Manipulation
Algebraic manipulation is the art of rearranging and transforming algebraic expressions in a series of logical steps. This can include a variety of techniques like distributing, factoring, combining like terms, and others. The goal is to make the expression simpler or to put it in a form that is easier to work with or understand.
In our exercise, after combining like terms, you are left with a clearer and more concise expression. The process of moving from \( -2 z^{3}+15 z+4 z^{3}+z^{2}-6 z^{2}+z \) to \(2 z^{3} - 5 z^{2} + 16 z\) involves adding and subtracting terms judiciously to reach a simplified form. The ability to manipulate algebraic expressions skillfully is essential for solving equations and understanding more advanced mathematics.
In our exercise, after combining like terms, you are left with a clearer and more concise expression. The process of moving from \( -2 z^{3}+15 z+4 z^{3}+z^{2}-6 z^{2}+z \) to \(2 z^{3} - 5 z^{2} + 16 z\) involves adding and subtracting terms judiciously to reach a simplified form. The ability to manipulate algebraic expressions skillfully is essential for solving equations and understanding more advanced mathematics.
Elementary Algebra
Elementary algebra is the most basic form of algebra taught to students. It constitutes the foundation of all advanced mathematical studies. The key topics in elementary algebra include the proper use of variables, handling equations and inequalities, and understanding functions.
The problem at hand showcases a basic operation in elementary algebra: simplification. Simplification is the process of reducing complexity in an algebraic expression, making it easier to interpret and solve. This act often involves processes like combining like terms and other algebraic manipulations. Mastering elementary algebra is a critical stepping stone for success in higher mathematics and many quantitative fields.
The problem at hand showcases a basic operation in elementary algebra: simplification. Simplification is the process of reducing complexity in an algebraic expression, making it easier to interpret and solve. This act often involves processes like combining like terms and other algebraic manipulations. Mastering elementary algebra is a critical stepping stone for success in higher mathematics and many quantitative fields.
Other exercises in this chapter
Problem 39
For the following problems, list, if any should appear, the common factors in the expressions. $$ 45 a b^{2}+9 b^{2} $$
View solution Problem 40
Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ 9 z=12 x-18 $$
View solution Problem 40
Use numerical evaluation on the equations. Astronomy (Kepler's law of planetary motion) \(P^{2}=k a^{3} .\) Find \(P^{2}\) if \(k=8\) and \(a=31\).
View solution Problem 40
For the following problems, perform the multiplications and combine any like terms. $$ -3(b+8) $$
View solution