Problem 9 - Elementary and Intermediate Algebra

Question

Write without parentheses. a. \(-\left(5 x^{2}-8 x+23\right)\) b. \(-\left(-5 y^{4}+3 y^{2}-7\right)\)

Step-by-Step Solution

Answer

a. \(-5x^2 + 8x - 23\); b. \(5y^4 - 3y^2 + 7\)

Step 1: Understand Negative Sign Distributive Property

When a negative sign is placed outside parentheses, it must be distributed to every term inside the parentheses. This means multiplying each term inside by -1.

Step 2: Apply Distribution in Part (a)

For expression a: 1. Multiply each term inside \(- \left(5x^2 - 8x + 23\right)\)by -1. 2. This results in \(-1 \times 5x^2 = -5x^2\), \(-1 \times (-8x) = 8x\), and \(-1 \times 23 = -23\).

Step 3: Write Result for Part (a)

The expression \(-\left(5x^2 - 8x + 23\right)\) becomes \(-5x^2 + 8x - 23\).

Step 4: Apply Distribution in Part (b)

For expression b:1. Multiply each term inside \(- \left(-5y^4 + 3y^2 - 7\right)\) by -1.2. This results in \(-1 \times (-5y^4) = 5y^4\), \(-1 \times 3y^2 = -3y^2\), and \(-1 \times (-7) = 7\).

Step 5: Write Result for Part (b)

The expression \(-\left(-5y^4 + 3y^2 - 7\right)\) becomes \(5y^4 - 3y^2 + 7\).

Key Concepts

Negative Sign

A negative sign can be like a hidden force that changes everything inside the parentheses in an equation. When you see a negative sign outside of parentheses, it means you need to apply it to every single term inside those parentheses. In simpler terms, you multiply each term by -1.

• If the term is positive, it becomes negative.
• If the term is already negative, it becomes positive because multiplying two negatives results in a positive.

This is a handy rule in algebra to change all the signs when needed, and it's essential when working with expressions inside parentheses.

Algebraic Expressions

Algebraic expressions are like mathematical phrases that can include numbers, variables, and operation symbols. Unlike equations, they do not include an equals sign. They are useful for simplifying mathematical problems and finding values.

Algebraic expressions can take various forms:

• Monomials: These have only one term, such as \(5x^2\) or -7.
• Binomials: These have exactly two terms, like \(x + 4\) or \(3y^2 - 5\).
• Trinomials: These contain three terms, such as \(2x^2 - 4x + 8\).

Remember that in Algebra, terms are separated by plus or minus signs. Each term can have constants, like \(23\), and variables, like \(x\), which are often raised to a power. Simplifying these algebraic expressions using rules like those of the negative sign helps solve many mathematical problems, making them manageable.

Parentheses Removal

Removing parentheses involves utilizing the distributive property, a useful tool in algebra that simplifies expressions for solving equations effectively. When you see parentheses in an expression with a factor outside, distributing means you need to multiply each term inside the parentheses by that factor.

Here's how to do it:

• Identify the term or factor outside the parentheses. This could be a number, variable, or in this case, a negative sign (-1).
• Multiply each term inside the parentheses by the factor outside.
• Write the new expression without parentheses, including each term with its new sign if a negative factor is involved.

This operation is an example of the distributive property which states: \( a(b + c) = ab + ac \). Similarly, \(- (b - c) = -b + c\) if the factor outside is -1. This rule ensures you correctly simplify and manipulate expressions to find solutions without any confusion.