Problem 27 Simplify each expression. $$\l... [FREE SOLUTION]

Chapter 13: Problem 27

Simplify each expression. $$\left(y^{3}+y+2\right)-4\left(y^{2}+y\right)$$

Short Answer

Expert verified

The expression simplifies to $y^3 - 4y^2 - 3y + 2$.

Step by step solution

Distribute the negative

The expression is $\left(y^{3}+y+2\right)-4\left(y^{2}+y\right)$. First, distribute the $-4$ across the terms inside the parentheses. This means you will multiply $-4$ by each term in $\left(y^{2}+y\right)$. This gives us $-4y^{2} - 4y$. So the expression becomes: \[y^{3} + y + 2 - 4y^{2} - 4y\]

Combine like terms

Now, combine the like terms from the expression $y^{3} + y + 2 - 4y^{2} - 4y$. Group the $y$ terms and other similar terms together. - The $y^3$ term stands alone.- Combine the $y$ terms: $y - 4y = -3y$.- There is only one constant term: $+2$.So, the expression simplifies to:\[y^3 - 4y^2 - 3y + 2\]

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Key Concepts

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

Combining Like Terms

Combining like terms is crucial for simplifying algebraic expressions. When we talk about "like terms," we're referring to terms that have the same variable raised to the same power. In the expression $y^3 + y + 2 - 4y^2 - 4y$, you can notice how terms are grouped based on their variables and exponents.

The term $y^3$ is unique because it is the only one with the variable $y$ raised to the third power.
The $y^2$ terms are represented only by $-4y^2$, making it stand alone as well.
We notice both $y$ and $-4y$ share the variable $y$ to the first power. By combining these, you perform the operation: $y - 4y$, which simplifies to $-3y$.

Grouping and adding/subtracting terms with the same variables is like filling the same container. Be sure each term belongs with like terms, and take care to apply any subtractive or additive operations accurately. This step is important for producing a simplified expression that is easier to interpret or solve.

Distributing Terms

Distributing terms means multiplying a term outside a parenthesis by each term inside the parenthesis. It is like spreading butter evenly on slices of bread. In the expression $\left(y^{3}+y+2\right)-4\left(y^{2}+y\right)$, notice the presence of the parenthesis and the $-4$ outside them.

Multiply $-4$ by each term inside the parenthesis $y^2 + y$.
Performing this multiplication step-by-step gives you $-4y^2$ from the first term and $-4y$ from the second term.

After distribution, the expression evolves to $y^{3}+y+2-4y^{2}-4y$. By ensuring we distribute accurately, the structure of the expression reflects the correct reduced form necessary for simplifying further.

Polynomials

Polynomials are expressions that contain several terms combined via addition or subtraction. They are fundamental building blocks in algebra, capable of describing a broad array of mathematical scenarios. In this simplification, we started with the expression $\left(y^{3}+y+2\right)-4\left(y^{2}+y\right)$, a polynomial already but presented in a slightly complex form.

The task was to simplify it by distributing and combining like terms, as shown in the previous steps.
After simplifying, we achieved a more straightforward polynomial: $y^3 - 4y^2 - 3y + 2$. This result illustrates how multiple terms of varying powers, from different sections of the expression, combine to form an organized sum.

Working with polynomials involves understanding how different terms interact through established rules, ensuring precise arithmetic while combining like terms, and carefully using distribution. Master these skills, and tackling polynomial expressions will become much more intuitive.

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91影视

Simplify each expression. $$\left(y^{3}+y+2\right)-4\left(y^{2}+y\right)$$

Short Answer

Step by step solution

Distribute the negative

Combine like terms

Key Concepts

Combining Like Terms

Distributing Terms

Polynomials

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