Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-12,\) write the sum or difference of the given polynomials in simplest form. $$ \left(4 x^{2}-3 x-7\right)+\left(3 x^{2}-2 x+3\right) $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(7x^2 - 5x - 4\).

Step 1: Identify Like Terms

In the expression \((4x^{2} - 3x - 7) + (3x^{2} - 2x + 3)\), identify the like terms which are those terms that have the same variable raised to the same power. Here, \(4x^2\) and \(3x^2\) are like terms because they involve \(x^2\), \(-3x\) and \(-2x\) are like terms as they involve \(x\), and \(-7\) and \(3\) are constant terms.

Step 2: Add Like Terms

Add the coefficients of like terms separately. - For \(x^2\) terms: \(4x^2 + 3x^2 = 7x^2\). - For \(x\) terms: \(-3x - 2x = -5x\). - For constant terms: \(-7 + 3 = -4\).

Step 3: Write the Simplified Polynomial

Combine the results from Step 2 to write the polynomial in its simplest form: \(7x^2 - 5x - 4\).

Key Concepts

Like Terms

In the world of polynomials, understanding "like terms" is crucial. Imagine trying to tidy up a cluttered room; finding like terms is a bit like sorting all the socks together. Like terms are terms in a polynomial that have the same variables raised to the same power. For example, in the polynomial expression \((4x^2 - 3x - 7) + (3x^2 - 2x + 3)\):

• \(4x^2\) and \(3x^2\) are like terms because 'x' is raised to the power of 2 in both, making them both of the category \(x^2\).
• \(-3x\) and \(-2x\) are like terms because both contain the variable 'x' raised to the first power.
• \(-7\) and \(3\) are constants, without any variable, making them like terms as well.

Identifying these terms makes the process of simplifying and adding polynomials much easier. It's as if you're putting all the similar items together before counting them.

Simplifying Polynomials

Once you've identified like terms in a polynomial, simplifying is just a step away. Simplifying polynomials is akin to combining items that belong together. By adding or subtracting the like terms, you can reduce the complexity of a polynomial expression. Take our expression:

• Add terms associated with \(x^2\): \(4x^2 + 3x^2\) results in \(7x^2\).
• Combine the linear terms: \(-3x - 2x\) simplifies to \(-5x\).
• For the constants, combine them: \(-7 + 3\) yields \(-4\).

After simplifying, the expression becomes \(7x^2 - 5x - 4\). Simplifying polynomials helps to make them easier to work with, whether you're graphing them or using them in further calculations.

Polynomial Expressions

Polynomial expressions are mathematical sentences involving variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They can be as simple as a monomial like \(4x\), or as complex as a polynomial such as \(4x^2 - 3x - 7\). These expressions can often be reorganized using operations like addition or subtraction to form new polynomials.When working with polynomials, it's helpful to:

• Recognize and work with like terms to simplify the expression.
• Understand the degree of the polynomial, which is the highest power of the variable in the expression. For instance, \(4x^2 - 3x - 7\) is a second-degree polynomial (the degree is 2).
• Use operations on polynomials—such as addition, subtraction, and multiplication—to derive new polynomial expressions.

By understanding the basic setup of polynomial expressions and how to manipulate them, solving polynomial addition problems like the one given becomes much more manageable.