Problem 36 - Algebra for College Students

Question

Subtract the polynomials using the vertical format. \(4 x^{2}-3 x-7\) from \(-x^{2}-6 x+9\)

Step-by-Step Solution

Answer

The result is \(-5x^2 - 3x + 16\).

Step 1: Write Each Polynomial in the Vertical Format

Align the polynomials one under the other with their like terms lined up. The first polynomial goes on top:\(\begin{array}{r}- x^{2} - 6x + 9\4x^{2} - 3x - 7\\hline\end{array}\)

Step 2: Change the Sign of the Subtracting Polynomial

Change the sign of each term in the polynomial you are subtracting from the other polynomial. Change the signs of: \(4x^2\) to \(-4x^2\), \(-3x\) to \(+3x\),\(-7\) to \(+7\).

Step 3: Add the Polynomials

Add the polynomials by combining like terms:\(\begin{array}{r}- x^{2} - 6x + 9\-(4x^{2}) + 3x + 7\\hline-5x^{2} - 3x + 16\\end{array}\)Combine \(-x^2 -4x^2 = -5x^2\), \(-6x + 3x = -3x\), \(9 + 7 = 16\).

Key Concepts

Vertical Format

When subtracting polynomials, using the vertical format can make the task much clearer and organized. Think of the vertical format as a way to stack the polynomials neatly, similar to how you would approach typical addition or subtraction problems in arithmetic.

To set up your polynomials in vertical format, follow these steps:

• Write the first polynomial (the one you are subtracting from) on top.
• Below it, align the second polynomial, making sure each corresponding term is directly under its like term from the first polynomial.

For example, when you have the polynomials \( -x^{2} - 6x + 9 \) and \( 4x^{2} - 3x - 7 \), place them as follows:
\(-x^{2} - 6x + 9\)
\( 4x^{2} - 3x - 7\)

By ensuring that the terms are properly aligned, you'll make the process of identifying like terms and performing operations on them much simpler.

Like Terms

Understanding like terms is crucial when working with polynomials, whether you’re adding or subtracting them. Like terms are terms that share the same variables raised to the same power. This allows them to be combined by simple addition or subtraction of their coefficients.

For instance, consider the polynomials \( -x^{2} - 6x + 9 \) and \( 4x^{2} - 3x - 7 \). Notice how they are aligned:

• The terms \( -x^{2} \) and \( 4x^{2} \) are like terms because they both involve \( x^{2} \).
• The terms \( -6x \) and \( -3x \) both involve \( x \) to the power of 1 and are thus like terms.
• Numbers like \( 9 \) and \( -7 \) without variables are constant terms and are like terms to each other.

Always ensure you only combine the coefficients of these like terms to avoid errors.

Changing Signs

When subtracting polynomials, one essential step is changing the signs of all terms in the polynomial that is being subtracted. This process turns a subtraction problem into an addition one, simplifying the operation considerably.

Here's how you go about changing signs:

• For \( 4x^{2} - 3x - 7 \), each term will have its sign flipped:
• The \( 4x^{2} \) becomes \( -4x^{2} \).
• The \(-3x\) becomes \( +3x \).
• The constant term \( -7 \) becomes \( +7 \).

Once the signs are changed, the problem is now set for addition, allowing us to effectively combine the newly aligned terms through addition:
\(-x^{2} - 6x + 9\)
+\((-4x^{2}) + 3x + 7\)

This step is crucial to avoid errors and ensure the operation is done correctly.