What is the quotient represented by the expression 8w2−20w−122w+1
for all values of w
where the expression is defined

The expression you provided is:

\[
\frac{8w^2 - 20w - 12}{2w + 1}
\]

This represents the quotient of a quadratic polynomial divided by a linear polynomial. We can simplify this using polynomial long division or synthetic division. Let's perform the polynomial division step by step.

### Step 1: Divide the leading term of the numerator by the leading term of the denominator

\[
\frac{8w^2}{2w} = 4w
\]

### Step 2: Multiply the entire denominator by 4w

\[
4w(2w + 1) = 8w^2 + 4w
\]

### Step 3: Subtract this result from the original numerator

\[
(8w^2 - 20w - 12) - (8w^2 + 4w) = -24w - 12
\]

### Step 4: Divide the leading term of the new polynomial by the leading term of the denominator

\[
\frac{-24w}{2w} = -12
\]

### Step 5: Multiply the entire denominator by -12

\[
-12(2w + 1) = -24w - 12
\]

### Step 6: Subtract this from the current polynomial

\[
(-24w - 12) - (-24w - 12) = 0
\]

### Final Result:
The division results in:

\[
4w - 12
\]

Thus, the quotient of the expression \(\frac{8w^2 - 20w - 12}{2w + 1}\) for all values of \(w\) (where the denominator is not zero) is:

\[
4w - 12
\]

### Domain Restriction:
Since the denominator is \(2w + 1\), the expression is undefined when:

\[
2w + 1 = 0 \implies w = -\frac{1}{2}
\]

Therefore, the quotient is defined for all values of \(w\) except \(w = -\frac{1}{2}\).

---

Would you like more details on this? Or any other questions?

Here are 5 related questions to explore:
1. How would you approach synthetic division for this problem?
2. What is the remainder when dividing these polynomials?
3. Can you factor the quadratic expression \(8w^2 - 20w - 12\)?
4. How do you interpret the result graphically?
5. What other methods can be used to divide polynomials?

**Tip:** Always check for values that make the denominator zero when simplifying rational expressions!

What is the quotient represented by the expression 8w^2 - 20w - 12 / 2w + 1 for all values of w where the expression is defined?

Learn how to find the quotient of the rational expression (8w^2 - 20w - 12) divided by (2w + 1). This step-by-step guide walks through polynomial long division and covers important algebraic concepts such as polynomial division and domain restrictions. Suitable for Grades 10-12.

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