Problem 2 - Algebra and Trigonometry

Question

To add polynomials, we add _______ terms. So \(\left(3 x^{2}+2 x+4\right)+\left(8 x^{2}-x+1\right)=\) _______.

Step-by-Step Solution

Answer

Add like terms to get \(11x^2 + x + 5\).

Step 1: Understand the Question

We are asked to add two polynomials: \(3x^2 + 2x + 4\) and \(8x^2 - x + 1\). To add polynomials, we need to combine like terms.

Step 2: Identify Like Terms

Like terms in polynomials have the same variable raised to the same power. For these polynomials, the like terms are: \(3x^2\) and \(8x^2\), \(2x\) and \(-x\), and the constant terms \(4\) and \(1\).

Step 3: Add the Like Terms

Add the coefficients of each pair of like terms:- For the \(x^2\) terms: \(3x^2 + 8x^2 = 11x^2\)- For the \(x\) terms: \(2x - x = x\)- For the constant terms: \(4 + 1 = 5\)

Step 4: Write the Resulting Polynomial

After adding the like terms, the resulting polynomial is \(11x^2 + x + 5\).

Key Concepts

Understanding Like Terms

In the world of polynomials, a critical concept to grasp is that of 'like terms'. Understanding this will make combining polynomials much easier. Like terms are terms that have exactly the same variable part. This means they should have the same variable raised to the same power. For example, in the expression \(3x^2 + 2x^2\), both terms are like terms because they each have the variable \(x\) raised to the power of 2.

• Must have identical variables
• Must have the same power

Identifying like terms is the first step in adding or subtracting polynomials since only like terms can be combined. This means no matter how similar the terms \(3x\) and \(3x^2\) look, they are not like terms because the variables are not raised to the same power.

The Role of Coefficients

Another fundamental aspect when dealing with polynomials is understanding the role of coefficients. Coefficients are the numerical parts of the terms. In the term \(3x^2\), the coefficient is \(3\).

• They multiply the variables
• Only coefficients of like terms are added together

Knowing the coefficients helps us add or subtract the polynomials effectively. For example, to add \(3x^2\) and \(8x^2\), we add the coefficients \(3\) and \(8\) to get \(11x^2\). This focuses the operation on numerical addition, as the variable part remains the same.

Combining Polynomials: Putting It All Together

Once we’ve identified the like terms and understood the coefficients, the next step is combining the polynomials. This involves adding the coefficients of like terms together to create a simplified polynomial.

• Identify like terms
• Add/subtract their coefficients
• Write the new expression

For instance, if we have the polynomials \(3x^2 + 2x + 4\) and \(8x^2 - x + 1\), we start by finding like terms: \(3x^2\) and \(8x^2\), \(2x\) and \(-x\), and constants \(4\) and \(1\).
By combining these, we get: - \(3x^2 + 8x^2 = 11x^2\) - \(2x - x = x\)- \(4 + 1 = 5\) Finally, we get the resulting polynomial \(11x^2 + x + 5\), which is a simplified combination of the original polynomials.