91Ó°ÊÓ

Polynomials are expressions that consist of variables and coefficients, connected by addition, subtraction, and multiplication. To add or subtract polynomials, you need to align and combine like terms. For example, in the exercise given, we are asked to subtract two polynomials:
\(j^{2}-k^{2}\) and \(j^{2}-8jk-5k^{2}\).
To begin, rewrite the expression by removing any parentheses. This gives us: \[ (j^2 - k^2) - (j^2 - 8jk - 5k^2) \]
With the parentheses removed, the expression becomes easier to manage. Always simplify expressions within parentheses before removing them to avoid errors.

Like terms in polynomials are terms that have the same variable(s) and the same power. In our exercise, the terms \(j^2\) and \(j^2\) are like terms because they both have the variable \(j\) raised to the power of 2. The terms \(k^2\) and \(5k^2\) are like terms because they both have \(k\) raised to the power of 2.
To combine like terms, add or subtract the coefficients:

• \( j^2 - j^2 = 0 \)
• \( 5k^2 - k^2 = 4k^2 \)
• \( 8jk \), which has no like terms, stays the same \

After simplifying, you combine these results to get the final simplified form.

When subtracting polynomials, it's crucial to distribute the negative sign across all terms in the polynomial being subtracted. This ensures each term is correctly negated. In our specific problem: \[ (j^2 - k^2) - (j^2 - 8jk - 5k^2) \] You must distribute the negative sign to each term inside the second set of parentheses: \[ j^2 - k^2 - j^2 + 8jk + 5k^2 \] This step changes the signs of each term in the second polynomial. The \( j^2 \) term becomes - \( j^2 \), - \( 8jk \) becomes + \( 8jk \), and - \( 5k^2 \) becomes + \( 5k^2 \).
Properly distributing the negative sign is an essential skill in polynomial operations. It helps you to avoid common mistakes and ensures accurate results.