Step-by-step Sample Problem
ion to divide the two polynomials.

in ( ) form? No
to multiply the divisor by to get a coefficient of 1 ?
are in the dividend? 5

Ask by Sandoval Ryan. in the United States

Mar 06,2025

Question

Step-by-step Sample Problem
ion to divide the two polynomials.

in ( ) form? No
to multiply the divisor by to get a coefficient of 1 ?
are in the dividend? 5

Ask by Sandoval Ryan. in the United States

Mar 06,2025

UpStudy AI · Accepted Answer

To divide the polynomial $$ -8w^4 - 34w^3 + 5w^2 - 10w + 36 $$ by $$ -2w - 9 $$, we will use polynomial long division. Let's break this down step by step.

### Step 1: Set Up the Division
We are dividing:
- **Dividend**: $$ -8w^4 - 34w^3 + 5w^2 - 10w + 36 $$
- **Divisor**: $$ -2w - 9 $$

### Step 2: Divide the Leading Terms
1. Divide the leading term of the dividend $$ -8w^4 $$ by the leading term of the divisor $$ -2w $$:
   $$
   \frac{-8w^4}{-2w} = 4w^3
   $$
   This is the first term of our quotient.

### Step 3: Multiply and Subtract
2. Multiply the entire divisor $$ -2w - 9 $$ by $$ 4w^3 $$:
   $$
   4w^3(-2w - 9) = -8w^4 - 36w^3
   $$
3. Subtract this result from the original dividend:
   $$
   (-8w^4 - 34w^3) - (-8w^4 - 36w^3) = 2w^3
   $$
   Now, our new polynomial is:
   $$
   2w^3 + 5w^2 - 10w + 36
   $$

### Step 4: Repeat the Process
4. Divide the leading term $$ 2w^3 $$ by $$ -2w $$:
   $$
   \frac{2w^3}{-2w} = -w^2
   $$
   This is the next term of our quotient.

5. Multiply the divisor by $$ -w^2 $$:
   $$
   -w^2(-2w - 9) = 2w^3 + 9w^2
   $$
6. Subtract this from the current polynomial:
   $$
   (2w^3 + 5w^2) - (2w^3 + 9w^2) = -4w^2
   $$
   Now, our new polynomial is:
   $$
   -4w^2 - 10w + 36
   $$

### Step 5: Continue Dividing
7. Divide $$ -4w^2 $$ by $$ -2w $$:
   $$
   \frac{-4w^2}{-2w} = 2w
   $$
   This is the next term of our quotient.

8. Multiply the divisor by $$ 2w $$:
   $$
   2w(-2w - 9) = -4w^2 - 18w
   $$
9. Subtract this from the current polynomial:
   $$
   (-4w^2 - 10w) - (-4w^2 - 18w) = 8w
   $$
   Now, our new polynomial is:
   $$
   8w + 36
   $$

### Step 6: Final Division
10. Divide $$ 8w $$ by $$ -2w $$:
    $$
    \frac{8w}{-2w} = -4
    $$
    This is the next term of our quotient.

11. Multiply the divisor by $$ -4 $$:
    $$
    -4(-2w - 9) = 8w + 36
    $$
12. Subtract this from the current polynomial:
    $$
    (8w + 36) - (8w + 36) = 0
    $$

### Conclusion
The division is complete, and there is no remainder. The quotient is:
$$
4w^3 - w^2 + 2w - 4
$$

### Answer to the Questions
1. The quotient in $$ X - r $$ form is $$ 4w^3 - w^2 + 2w - 4 $$.
2. The coefficient of $$ 1 $$ is not needed in this case since we have already divided completely.
3. The number of terms in the dividend is $$ 5 $$.

If you have any further questions or need additional assistance, feel free to ask!
To divide $$ -8w^4 - 34w^3 + 5w^2 - 10w + 36 $$ by $$ -2w - 9 $$, perform polynomial long division step by step. The quotient is $$ 4w^3 - w^2 + 2w - 4 $$, and there are 5 terms in the dividend.

Step-by-step Sample Problem
ion to divide the two polynomials.

in ( ) form? No
to multiply the divisor by to get a coefficient of 1 ?
are in the dividend? 5

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Step-by-step Sample Problem ion to divide the two polynomials. in ( ) form? No to multiply the divisor by to get a coefficient of 1 ? are in the dividend? 5

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Step-by-step Sample Problem
ion to divide the two polynomials.

in ( ) form? No
to multiply the divisor by to get a coefficient of 1 ?
are in the dividend? 5