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Long division is a method used to divide polynomials, much like how you divide numbers. Here's a simplified rundown:
- **Step 1:** Set up your division. The dividend (what you're dividing) goes under the long division symbol, and the divisor (what you're dividing by) goes outside.
- **Step 2:** Divide the leading terms. Look at the first term of the dividend and the first term of the divisor. Divide those to get the first term of your answer.
- **Step 3:** Multiply and subtract. Multiply your answer term by the entire divisor, then subtract that result from the dividend.
- **Step 4:** Repeat. Continue dividing, multiplying, and subtracting until you can't divide anymore.
This process helps break down complex polynomial divisions into simpler, manageable steps. It is crucial to keep your work neat and follow each step carefully, so you don't miss any important details.

Algebraic expressions are a combination of numbers, variables (like w), and operators (like +, -, *). In this exercise, our algebraic expression is a polynomial: \[ w^{2}+4w-21 \]. Polynomials are sums of terms with variables raised to whole number exponents and can have constants (like -21).
When dealing with expressions, keep these points in mind:
- **Terms:** Separate pieces of the expression, like \( w^2 \), \( 4w \), and \( -21 \).
- **Coefficients:** Numbers in front of variables, like 4 in \( 4w \).
- **Variables and Exponents:** Letters representing unknown values, with their respective exponents. Here, 'w' is the variable, and 2 is its exponent in \( w^2 \).
Understanding these components is crucial in dividing and working with polynomials.

The Remainder Theorem tells us something useful about polynomials. If you divide a polynomial \( P(x) \) by \( x - c \), the remainder of that division is \( P(c) \).
In our exercise, after dividing \( w^2 + 4w - 21 \) by \( w + 3 \), we ended up with a remainder of -24. This aligns perfectly with the theorem since:
- We divided by \( w + 3 \), which is \( w - (-3) \).
This means the remainder should be \( P(-3) \), validating our solution.
The theorem helps simplify checking our work and understanding the behavior of polynomials when divided.