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

To find the quotient represented by the expression:

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

we'll use polynomial long division to divide the quadratic expression \( 8w^2 - 20w - 12 \) by the linear expression \( 2w + 1 \).

### Step 1: Divide the leading terms
First, divide the leading term of the numerator (\( 8w^2 \)) by the leading term of the denominator (\( 2w \)):

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

This gives the first term of the quotient, \( 4w \).

### Step 2: Multiply and subtract
Next, multiply \( 4w \) by the entire denominator \( 2w + 1 \):

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

Now subtract this from the original numerator:

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

### Step 3: Divide the new leading term
Now divide the leading term of the new expression (\( -24w \)) by the leading term of the denominator (\( 2w \)):

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

This gives the next term of the quotient, \( -12 \).

### Step 4: Multiply and subtract
Now multiply \( -12 \) by the denominator \( 2w + 1 \):

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

Now subtract this from the remaining expression:

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

There is no remainder, so the quotient is:

\[
4w - 12
\]

### Final Answer:
The quotient is:

\[
4w - 12
\]

where the expression is defined for all values of \( w \) except \( w = -\frac{1}{2} \), because \( 2w + 1 = 0 \) when \( w = -\frac{1}{2} \).

Would you like a detailed explanation of any of these steps or have any further questions?

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Here are 5 related questions to expand on this topic:

1. What is the process of polynomial long division in general?
2. Can you divide other polynomials by binomials using synthetic division?
3. How do you find the remainder in polynomial division when there is one?
4. What happens if the divisor has a degree higher than the dividend?
5. How do you handle cases where the divisor is a quadratic?

**Tip:** When dividing polynomials, always ensure that the degrees of terms are aligned and ordered from highest to lowest for easy division.

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

This detailed solution walks through how to find the quotient of 8w^2 − 20w − 12 divided by 2w + 1 using polynomial long division. The problem illustrates key algebraic concepts like polynomial division and rational expressions, suitable for Grades 10-12.

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