Problem 16 - Algebra for College Students

Question

Add the given polynomials. \(6 x^{2}+8 x+4\) and \(-7 x^{2}-7 x-10\)

Step-by-Step Solution

Answer

The sum of the polynomials is \(-x^2 + x - 6\).

Step 1: Align the Polynomials

Write the polynomials you need to add vertically, aligning like terms (those with the same degree). Place the polynomial with the highest degree term on top for ease of alignment:\[ \begin{align*} &6x^2 + 8x + 4 \ &-7x^2 - 7x - 10 \ \end{align*} \]

Step 2: Add the Like Terms

Add the coefficients of the polynomial terms that have the same degree:- For the \(x^2\) terms: \(6x^2 + (-7x^2) = -1x^2\)- For the \(x\) terms: \(8x + (-7x) = 1x\)- For the constant terms: \(4 + (-10) = -6\)So the result is:\[-1x^2 + 1x - 6\]

Step 3: Simplify the Expression

Write the polynomial as a simple expression:\[-x^2 + x - 6\]

Key Concepts

Like Terms

When working with polynomials, like terms are pivotal in simplifying expressions. Essentially, like terms are terms that have the exact same variables raised to the same powers. For instance, in polynomial operations, any terms that contain the variable \(x^2\) are like terms with one another. This also applies to terms such as \(x\) and any constant values (those without variables) respectively. Recognizing and aligning like terms makes the process of adding or subtracting polynomials straightforward.

To effectively identify and work with like terms, remember:

• Like terms share the same variable and the same power.
• They can be combined by adding or subtracting their coefficients while the variable part remains unchanged.
• This consolidation simplifies polynomial expressions, which is key for clarity and ease of computation.

Polynomial Expressions

Polynomial expressions are mathematical sentences involving a sum of powers in one or more variables multiplied by coefficients. They contain terms, which each consist of a coefficient (a number) and a variable with an exponent (e.g., \(5x^3\)). Each term stands alone but contributes to the polynomial expression.

Key characteristics of polynomial expressions include:

• The highest power or degree of the variable determines the degree of the polynomial. For instance, the degree of \(6x^2 + 8x + 4\) is 2.
• Coefficients are the numerical parts in front of the variables, which determine how many times the variable terms contribute to the expression.
• Polynomials can be rearranged by rearranging their terms, generally listed from the highest to lowest degree for standard form.

Simplifying Polynomials

Simplification of polynomials involves combining like terms and ensuring the polynomial is expressed in its simplest form. The aim is to make the expression cleaner and easier to interpret or solve in further equations.

Here’s how to simplify polynomial expressions effectively:

• Combine like terms: Group the terms with the same variable and degree, then add or subtract their coefficients.
• Order the polynomial: After combining like terms, write the expression starting with the highest power term descending to the constant term.
• Check for further simplifications: Ensure there are no more like terms left uncombined and that factors are simplified if possible.

For example, in the polynomial \(-x^2 + x - 6\) as derived from the addition of two polynomials, it’s evident that all like terms have been properly combined, and the polynomial is presented in decreasing order of power.