91Ó°ÊÓ

Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. In our exercise, we deal with simple polynomial expressions in the variable x.

A polynomial expression could be something like:

• \(ax^n + bx^{n-1} + \text{...} + zx^0 \)

where each term consists of a coefficient (like a, b, z) and a variable (like x) raised to a non-negative integer exponent.

For example, two polynomial expressions from the exercise are: \(9x^2 - 6x - 20\) and \(5x^2 - x + 12\).

When working with polynomials, especially in subtraction, grasping their structure is crucial. Each term in the polynomial is organized based on its degree (the exponent). This aids in combining terms efficiently in later steps.

When subtracting polynomial expressions, it is essential to distribute the negative sign correctly. Distributing means multiplying the negative sign across all terms of the polynomial you are subtracting.

Let's look at the subtraction expression from our exercise: \( (9x^2 - 6x - 20) - (5x^2 - x + 12) \).

We distribute the negative sign to each term in the second polynomial:

• \((5x^2)\) becomes \((-5x^2)\)
• \((-x)\) becomes \((+x)\)
• \((+12)\) becomes \((-12)\)

After distribution, the subtraction problem is now simplified to: \(9x^2 - 6x - 20 - 5x^2 + x - 12\).

Correctly distributing the negative sign ensures each term from the second polynomial is properly accounted for in subtraction, preventing common mistakes.

Combining like terms is the final crucial step in polynomial subtraction. Like terms are terms that have the same variable raised to the same power. Only these can be added or subtracted from each other.

From our adjusted polynomial expression: \(9x^2 - 6x - 20 - 5x^2 + x - 12\), we group and combine the like terms:

• Terms containing \(x^2\): \(9x^2 - 5x^2\)
• Terms containing \(x\): \(-6x + x\)
• Constant terms: \(-20 - 12\)

By combining these, we execute the following calculations:

• \((9x^2 - 5x^2) = 4x^2\)
• \((-6x + x) = -5x\)
• \((-20 - 12) = -32\)

Thus, the final simplified result of the polynomial subtraction is: \(4x^2 - 5x - 32\).

Remembering to group and combine like terms efficiently gets you to the answer quickly and correctly.