Question

Divide​ f(w) by​ d(w), and write a summary statement in polynomial form and fraction form. f(w)=-4w ^3+4w ^2-5; d(w)=w-3 The summary statement written in polynomial form is -4w^3+4w^2-5= (Simplify your answer. Type exponential notation with positive​ exponents.)

Answer 1

The simplified form of
`\( (f(w))/(d(w)) \)\\` is
`\( -4w^2 + 8w + 19 - (16w - 24)/(w - 3) \)`. In polynomial form:
`\( -4w^2 + 8w + 19 \)`, fraction form:
`\( (-4w^2 + 8w + 19)/(w - 3) \).`

To find the simplified form of
`\( (f(w))/(d(w)) \)`, you need to perform polynomial division. The division involves dividing the leading term of f(w) by the leading term of d(w) and then multiplying d(w) by the result to subtract from f(w) . Repeat this process until the degree of the remainder is less than the degree of d(w) .

Let's perform the polynomial division:

`\[ f(w) = -4w^3 + 4w^2 - 5 \]`

`\[ d(w) = w - 3 \]`

1. Divide the leading term of f(w) by the leading term of d(w):

`\[ -4w^3 / (w) = -4w^2 \]`

2. Multiply d(w) by the result and subtract from f(w) :

(-4w^2)(w - 3)

-4w^3 + 12w^2

- (-4w^3 + 4w^2 - 5)

8w^2 - 5

3. Repeat the process with the new polynomial 8w^2 - 5 :

`\[ (8w^2 - 5) / (w - 3) = 8w + 19 \]`

4. Multiply d(w) by the result and subtract from 8w^2 - 5 :

(8w + 19)(w - 3)

8w^2 - 16w + 19

- (8w^2 - 5)

-16w + 24

Now, we have a linear polynomial -16w + 24 , and the degree of the remainder is less than the degree of d(w) (which is w - 3 ). Therefore, the final result is:

`\[ (f(w))/(d(w)) = -4w^2 + 8w + 19 - (16w - 24)/(w - 3) \]`

In polynomial form:

`\[ -4w^2 + 8w + 19 - (16w - 24)/(w - 3) \]`

In fraction form:

`\[ (-4w^2 + 8w + 19)/(w - 3) - (16w - 24)/(w - 3) \]`

The probable question maybe:

If f(w) = -4w^3 + 4w^2 - 5 and d(w) = w - 3 what is the simplified form of
`\( (f(w))/(d(w)) \)` and how would you express it in both polynomial form and fraction form?