Polynomials are algebraic expressions that consist of variables raised to whole number exponents, and manipulating them through addition and subtraction is a fundamental skill in algebra. The key to adding or subtracting polynomials lies in combining like terms, which are terms that have the same variable raised to the same power.

To add polynomials, you align the like terms and perform the addition. For example, consider the polynomials \(5x^2 + 2x + 3\) and \(x^2 + 7x + 8\). You would group the \(x^2\) terms, the \(x\) terms, and the constant terms separately:

• For \(x^2\) terms: \(5x^2 + x^2 = 6x^2\)
• For \(x\) terms: \(2x + 7x = 9x\)
• For constant terms: \(3 + 8 = 11\)

Thus, the sum of these polynomials is \(6x^2 + 9x + 11\), which is expressed in standard form, where the terms are arranged in decreasing order of their exponents.

Subtracting polynomials follows a similar process, but it requires careful attention to the distribution of negative signs. For instance, in the expression \(3x^2 + 2x + 4 - (5x + 10 - x^2)\), you first distribute the negative sign across the terms in the parentheses:

• Distributing gives: \(3x^2 + 2x + 4 - 5x - 10 + x^2\)

Next, combine the like terms:

• For \(x^2\) terms: \(3x^2 + x^2 = 4x^2\)
• For \(x\) terms: \(2x - 5x = -3x\)
• For constant terms: \(4 - 10 = -6\)

The result is \(4x^2 - 3x - 6\), also in standard form. It is crucial to be meticulous during subtraction to avoid errors, especially when distributing negative signs. Mastering these operations will enhance your ability to work with polynomials effectively.