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Chapter 10: Problem 42

Use a horizontal format to add or subtract. $$12-\left(y^{3}+4\right)$$

Short Answer

Expert verified
The simplified form of the given expression is \(8 - y^{3}\).

Step by step solution

01

Identify the Operation

In this problem, the operation to be performed is subtraction. Identify the expression within the parentheses, which is \(y^{3} + 4\).
02

Distribute Sign Across Brackets

When the minus sign is distributed across the parentheses, the problem simplifies to \(12 - y^{3} - 4\). Keep in mind that the plus between \(y^{3}\) and 4 becomes a minus when it goes outside the parentheses.
03

Perform the subtraction

Now, subtract 4 from 12, leaving the equation as \(8 - y^{3}\).

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Key Concepts

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

subtraction
Subtraction is a fundamental mathematical operation where you take one number away from another. In the given expression, subtraction is used to remove the expression inside the parentheses from 12. Recognizing the subtraction sign is crucial as it indicates how quantities are being combined in an equation.

When we say subtract, we mean taking away the value of one expression from another. For example, in the original exercise, we have to subtract the entire expression \(y^{3} + 4\) from 12. This involves two core actions: the arithmetic operation itself and handling of terms within the brackets.
distributive property
The distributive property allows us to multiply a single term across a set of terms inside parentheses. However, in the context of subtraction, it helps to apply the reverse operation: spreading a subtraction sign across terms within brackets.

In the problem, once we notice a subtraction outside the parentheses, this sign should be distributed to each term inside. So, starting with \(12 - (y^{3} + 4)\), you apply the minus to both \(y^{3}\) and 4, resulting in \(12 - y^{3} - 4\). This step ensures that each part of the expression within the parentheses is accurately subtracted from the initial number.
simplification
Simplification is the process of reducing an algebraic expression to its simplest form. After distributing the subtraction sign across the terms within parentheses, simplification helps to clean up the equation.

In our example, once we have the expression \(12 - y^{3} - 4\), the next task is to simplify this by combining like terms. Subtraction of numbers is straightforward, leading us from 12 to 8 after subtracting 4. At the end, the simplified expression becomes \(8 - y^{3}\), which is much easier to interpret and use in further calculations.
polynomial operations
Polynomial operations involve addition, subtraction, multiplication, and division of polynomial expressions. In the problem, the operation focused on is the subtraction of a polynomial, \(y^{3} + 4\), from a constant, 12.

Understanding polynomial operations requires recognizing terms and coefficients. Here, \(y^{3}\) is a term with a variable, while 4 is a constant term without variables. The result of subtracting these polynomial quantities from a constant helps form a new polynomial. Thus, the final expression, \(8 - y^{3}\), combines these operations, showcasing the interaction between constants and variable expressions in algebra.

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Most popular questions from this chapter

The safe working load \(S\) (in tons) for a wire rope is a function of \(D\), the diameter of the rope in inches. Safe working load model for wire rope: \(4 \cdot D^{2}=S\) When determining the safe working load \(S\) of a rope that is old or worn, decrease \(S\) by \(50 \% .\) Write a model for \(S\) when using an old wire rope. What diameter of old wire rope do you need to safely lift a 9 -ton load?

Use factoring to solve the equation. Use a graphing calculator to check your solution if you wish. $$x^{2}-\frac{5}{3} x+\frac{25}{36}=0$$

Sketch the graph of the inequality. \(y-4 x \leq 10\)

Use factoring to solve the equation. Use a graphing calculator to check your solution if you wish. $$\frac{1}{5} x^{2}-2 x+5=0$$

Find the product. $$(2 a-7)(2 a+7)$$
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