Problem 53 - Algebra A Combined Function

Question

Subtract \(\left(4 y^{2}-6 y-3\right)\) from the sum of \(\left(8 y^{2}+7\right)\) and \((6 y+9)\)

Step-by-Step Solution

Answer

The result of the operation is \(4y^2 + 12y + 19\).

Step 1: Compute the Sum

First, find the sum of the two given expressions: \((8y^2 + 7)\) and \((6y + 9)\). Combine similar terms to form a single expression:\[ (8y^2 + 7) + (6y + 9) = 8y^2 + 6y + 16 \]

Step 2: Setup for Subtraction

Write down the expression that needs to be subtracted from the sum. We need to subtract \((4y^2 - 6y - 3)\) from the expression obtained in Step 1:\[ 8y^2 + 6y + 16 - (4y^2 - 6y - 3) \]

Step 3: Distribute the Negative Sign

Distribute the negative sign across the terms in the expression being subtracted:\[ 8y^2 + 6y + 16 - 4y^2 + 6y + 3 \]

Step 4: Simplify the Expression

Combine like terms from the resulting expression:- Combine \(8y^2\) and \(-4y^2\) to get \(4y^2\).- Combine \(6y\) and \(6y\) to get \(12y\).- Combine \(16\) and \(3\) to get \(19\).Thus, the simplified expression is:\[ 4y^2 + 12y + 19 \]

Key Concepts

Combining Like Terms

Combining like terms is a crucial skill in algebra that simplifies expressions. When dealing with polynomial expressions, similar terms can be combined to make calculations easier.
This process involves looking for terms that share the same variables raised to identical powers.
For example, in the expression \(8y^2 + 6y + 16 - 4y^2 + 6y + 3\), you will notice like terms:

• Terms with \(y^2\): \(8y^2\) and \(-4y^2\).
• Terms with \(y\): \(6y\) and another \(6y\).
• Constant numbers: \(16\) and \(3\).

Combining these terms gives more manageable expressions. For terms with \(y^2\), we add \(8y^2\) and \(-4y^2\) to get \(4y^2\).
The terms with \(y\) sum up to \(12y\), and the constants add up to \(19\).
This results in a simplified expression: \(4y^2 + 12y + 19\).
By combining like terms, a complex expression is effectively reduced, making it easier to work with and understand.

Polynomial Expressions

A polynomial expression is an algebraic expression involving one or more terms, consisting of variables and coefficients.
Each term in a polynomial is made up of a coefficient multiplied by a variable raised to a nonnegative integer power.
Consider the expression \(8y^2 + 7\). It is a polynomial expression, specifically a quadratic, because the highest power of the variable \(y\) is 2. Polynomials can have:

• Constant terms: numbers with no variables, like \(7\) or \(3\).
• Linear terms: terms with the variable raised to the first power, such as \(6y\).
• Quadratic terms: terms with the variable squared, like \(8y^2\).

Understanding polynomial expressions is fundamental in algebra because they form the basis of many equations and functions.
You can perform various operations, like addition, subtraction, and multiplication on polynomials.
In this exercise, understanding how to add and subtract polynomial expressions is essential in simplifying them.

Algebraic Simplification

Algebraic simplification is the process of transforming an expression into a simpler or more manageable form.
This usually involves operations like combining like terms, distributing multiplication over addition or subtraction, and eliminating unnecessary terms.In the exercise provided, the algebraic simplification process included subtracting one polynomial expression from another.
Initially, we had to subtract \((4y^2 - 6y - 3)\) from the sum of \((8y^2 + 7) + (6y + 9)\).
The key steps in algebraic simplification included:

• Calculating the initial sum of two polynomial expressions.
• Distributing the negative sign across the polynomial being subtracted.
• Combining all resulting like terms to produce the simplified expression.

This step-by-step process helps to transform a potentially complicated expression into its most straightforward form, making further algebraic operations much easier.
Simplification not only makes expressions easier to read but also paves the way for solving equations efficiently.