Problem 35 - Algebra for College Students

Question

Subtract the polynomials using the vertical format. \(2 x^{2}+x+6\) from \(4 x^{2}-x-2\)

Step-by-Step Solution

Answer

The result is \(2x^2 - 2x - 8\).

Step 1: Write the First Polynomial

First, write the polynomial you are subtracting from: \[4x^2 - x - 2\]

Step 2: Write the Second Polynomial Below the First

Write the polynomial you are subtracting below the first polynomial, aligning like terms:\[\begin{array}{r} 4x^2 - x - 2 \2x^2 + x + 6 \\hline\end{array}\]

Step 3: Change the Signs of the Second Polynomial

Change the signs of each term in the second polynomial:\[\begin{array}{r} 4x^2 - x - 2 \-2x^2 - x - 6 \\hline\end{array}\]

Step 4: Subtract the Polynomials

Subtract each column to find the difference for each term:- Subtract the \(x^2\) terms: \(4x^2 - 2x^2 = 2x^2\)- Subtract the \(x\) terms: \(-1x - 1x = -2x\)- Subtract the constant terms: \(-2 - 6 = -8\)The resulting polynomial is:\[2x^2 - 2x - 8\]

Step 5: Write the Result

The final result after subtracting the polynomials is:\[2x^2 - 2x - 8\]

Key Concepts

Vertical Format

When it comes to polynomial subtraction, using the vertical format can make the process much clearer and more organized. This method is similar to how we traditionally subtract numbers, by writing them one below the other.
In the vertical format, each term of a polynomial is aligned based on its degree. This means that terms with the same powers of the variable are placed in the same column. For example, for the polynomials \(4x^2 - x - 2\) and \(2x^2 + x + 6\), the \(x^2\) terms, \(x\) terms, and constant terms are aligned together.
By arranging the polynomials this way, it becomes easy for us to visualize and manage the subtraction step by step, tackling each column one at a time.

Aligning Like Terms

Aligning like terms is crucial in polynomial subtraction to ensure the subtraction operation is carried out correctly. Like terms are terms that have the same variable raised to the same power.
In our problem, the polynomials are \(4x^2 - x - 2\) and \(2x^2 + x + 6\). By aligning them vertically, we can see that:

• The \(4x^2\) term is placed directly above the \(2x^2\) term.
• The \(-x\) term is aligned with the \(x\) term.
• The constant \(-2\) is placed above the constant \(6\).

Aligning the terms in this manner allows us to focus on subtracting the coefficients of like terms. This method helps avoid mistakes and makes the subtraction process smoother.

Changing Signs

In polynomial subtraction, changing the signs of the polynomial being subtracted is essential. When we subtract, what we are doing is adding the additive inverse of a number or expression.
To obtain the additive inverse, we change the signs of all terms in the second polynomial. For the polynomial \(2x^2 + x + 6\), the signs of the terms change to \(-2x^2\), \(-x\), and \(-6\).
This sign change transforms the subtraction problem into an addition problem, allowing us to add together the aligned terms, which simplifies the process.
Sign change needs to be carefully applied to each term to ensure the final result is correct.

Subtracting Coefficients

The next step involves subtracting the coefficients of the like terms. Think of this as performing simple arithmetic on the numbers in front of the variables, called coefficients.
For our example:

• The coefficients of \(x^2\) are 4 and -2: \(4 - 2 = 2\).
• The coefficients of \(x\) are -1 and -1: \(-1 - 1 = -2\).
• The constant terms are \(-2\) and \(-6\): \(-2 - 6 = -8\).

Once the coefficients have been subtracted, you can recombine them with their corresponding variables to form the new polynomial.
The final result is \(2x^2 - 2x - 8\). This systematic approach ensures accuracy and helps build confidence in handling polynomial operations.