Problem 11 - Elementary and Intermediate Algebra

Question

Fill in the blanks to add (subtract) the polynomials. $$ \left(6 x^{2}+2 x+3\right)+\left(4 x^{2}-7 x+1\right) $$ $$ \begin{aligned} &=\left(6 x^{2}+\quad\right)+(\quad-7 x)+(3+\quad)\\\ &=\quad-5 x+ \end{aligned} $$

Step-by-Step Solution

Answer

The summed polynomial is \(10x^2 - 5x + 4\).

Step 1: Combine Like Terms for Each Polynomial Degree

Identify the like terms in the two polynomials. The expression given is \((6x^2 + 2x + 3) + (4x^2 - 7x + 1)\). You need to combine the terms with similar degrees. Specifically, \(6x^2\) and \(4x^2\) are the quadratic terms, \(2x\) and \(-7x\) are the linear terms, and \(3\) and \(1\) are the constant terms.

Step 2: Add the Quadratic Terms

Add the quadratic terms \(6x^2\) and \(4x^2\) together. The calculation is \(6x^2 + 4x^2 = 10x^2\).

Step 3: Add the Linear Terms

Combine the linear terms \(2x\) and \(-7x\). The calculation is \(2x - 7x = -5x\).

Step 4: Add the Constant Terms

Add the constant terms \(3\) and \(1\). The calculation is \(3 + 1 = 4\).

Step 5: Write the Final Combined Polynomial

Combine all the results from the previous steps: \(10x^2\) from the quadratic terms, \(-5x\) from the linear terms, and \(+4\) from the constant terms. So, the final polynomial is \(10x^2 - 5x + 4\).

Key Concepts

Quadratic Terms in Polynomial Addition

Quadratic terms are those parts of a polynomial that include the variable squared, expressed as \( ax^2 \). In our exercise, the quadratic terms are \( 6x^2 \) and \( 4x^2 \). When adding or subtracting polynomials, we need to combine these terms separately from others because they share the same degree.
To add quadratic terms:

• Align terms with the same degree.
• Add the coefficients, maintaining the squared variable intact.

In this example, the addition \( 6x^2 + 4x^2 \) results in \( 10x^2 \). This calculation involves adding the coefficients 6 and 4, while the variable part \( x^2 \) remains unchanged. This ensures that the simplified result retains the degree of the quadratic term, making it essential for maintaining the polynomial's structure.

Linear Terms and Their Role

Linear terms in a polynomial are expressed in the form \( bx \). These are terms where the variable appears to the first power, like \( 2x \) and \(-7x \) in our current exercise. Although they involve the variable x, they don't carry the squared degree that quadratic terms do.
When combining linear terms:

• Focus on adding or subtracting the coefficients of \( x \).
• Remember that the degree, which is 1 in linear terms, stays constant during addition or subtraction.

Combining \( 2x - 7x \) involves a straight subtraction of the coefficients, yielding \( -5x \). The process shows how linear terms combine by focusing on their coefficients while the variable \( x \) continues to be linear.

Understanding Constant Terms

Constant terms are the numbers without variables in polynomial expressions, such as \( 3 \) and \( 1 \) in our polynomial. They represent the polynomial's fixed value component, which doesn't change with the input value of the variable.
To add constant terms:

• Simply add the numbers together.
• No variables are involved, making the process straightforward.

Adding the constants \( 3 + 1 \) results in \( 4 \). Constant terms often provide a baseline for the polynomial's expression. When completing polynomial addition, ensure that these terms are accurately summed to reflect any changes resulting from the addition or subtraction process.