Problem 50 - Algebra A Combined Function

Question

Add \(\left(-3 x^{2}-5 x+2\right)\) and \(\left(x^{2}-6 x+9\right)\)

Step-by-Step Solution

Answer

The sum is \(-2x^{2} - 11x + 11\).

Step 1: Identify and Write Down the Expressions

The first step is to clearly identify and write down the polynomial expressions that need to be added: \(-3x^{2}-5x+2\) and \(x^{2}-6x+9\).

Step 2: Align Like Terms

Next, align terms of the same degree, one under the other, so they can be easily combined:\[\begin{array}{cccc}-3x^{2} & -5x & +2 & \x^{2} & -6x & +9 & \\hline\end{array}\]

Step 3: Add the Like Terms

Now we add the like terms column by column:1. Add the quadratic terms: \(-3x^{2} + x^{2} = -2x^{2}\)2. Add the linear terms: \(-5x - 6x = -11x\)3. Add the constant terms: \(+2 + 9 = +11\).The result is \(-2x^{2} - 11x + 11\).

Step 4: Combine and Write the Result

Lastly, combine the results to write the final summed polynomial expression:\(-2x^{2} - 11x + 11\).

Key Concepts

Understanding Like Terms

When working with polynomials, like terms are crucial to understand. Like terms are terms in an expression that have the same variable raised to the same power. This uniformity allows them to be combined through addition or subtraction. For example, in the expression \(-3x^2 - 5x + 2\), the terms \(-3x^2\) and \(x^2\) from a separate expression, can be considered like terms because both involve \(x\) raised to the power of 2.
In the process of polynomial addition, identifying like terms simplifies the task considerably. This is because you can only add coefficients of terms that belong to the same 'family' of like terms. For instance, adding the expressions \((-3x^2 - 5x + 2)\) and \((x^2 - 6x + 9)\), requires detecting and aligning the like terms:

• Quadratic terms: \(-3x^2\) and \(x^2\)
• Linear terms: \(-5x\) and \(-6x\)
• Constant terms: \(+2\) and \(+9\)

Recognizing like terms helps streamline the addition process, ensuring that each distinct category of terms is treated correctly.

Examining a Quadratic Expression

A quadratic expression is a type of polynomial that includes a term with a variable raised to the power of two. Such expressions are fundamental in algebra and appear frequently across various mathematical applications.
Quadratic expressions typically follow the form \(ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants, and \(a eq 0\). For instance, in the expression \(-3x^2 - 5x + 2\), you can see:

• The quadratic term: \(-3x^2\)
• The linear term: \(-5x\)
• The constant term: \(+2\)

The focal point of quadratic expressions is the term with \(x^2\), which often affects the expression's graph with a distinctive parabolic shape. Understanding the components of a quadratic expression is essential when adding or simplifying polynomials, as it ensures each term is properly considered in the process.

Combining Polynomials

Combining polynomials involves one of the fundamental operations in algebra: addition. The goal is to simplify multiple polynomial expressions into a single, unified polynomial.
When adding polynomials like \((-3x^2 - 5x + 2)\) and \((x^2 - 6x + 9)\), the essential step is to align and add the like terms. This approach ensures that the resultant expression is as simplified as possible:

• Quadratic terms: Combine \(-3x^2\) and \(x^2\) to get \(-2x^2\)
• Linear terms: Combine \(-5x\) and \(-6x\) to get \(-11x\)
• Constant terms: Combine \(+2\) and \(+9\) to get \(+11\)

The final result is a new polynomial: \(-2x^2 - 11x + 11\). Adding polynomials is a straightforward process, given that you correctly align and add the like terms, yielding a concise expression that is easier to work with.