4. Find $$ f(x) \div g(x) $$ $$ \begin{array}{l}\text { a. } f(x)=4 x^{5}+12 x^{4} \div x^{3}+4 \ g(x)=x-1 \ \text { b. } f(x)=-6+4 x+5 x^{2} \ g(x)=x^{2}\end{array} $$

Question

UpStudy AI · Accepted Answer

To find $$ f(x) \div g(x) $$ for the given functions, we will perform polynomial long division for each case.

### Part a
Given:
$$
f(x) = 4x^5 + 12x^4 \div x^3 + 4
$$
and
$$
g(x) = x - 1
$$

First, we need to clarify $$ f(x) $$. It seems there is a misunderstanding in the expression. Let's assume $$ f(x) = 4x^5 + 12x^4 + 4 $$ (the division by $$ x^3 $$ is not clear).

Now, we will divide $$ f(x) $$ by $$ g(x) $$.

1. **Set up the division**:
   $$
   \text{Divide } 4x^5 + 12x^4 + 4 \text{ by } x - 1.
   $$

2. **Perform the long division**:
   - Divide the leading term: $$ \frac{4x^5}{x} = 4x^4 $$.
   - Multiply $$ 4x^4 $$ by $$ x - 1 $$: 
     $$
     4x^4(x - 1) = 4x^5 - 4x^4.
     $$
   - Subtract:
     $$
     (4x^5 + 12x^4) - (4x^5 - 4x^4) = 12x^4 + 4x^4 = 16x^4.
     $$
   - Bring down the $$ +4 $$:
     $$
     16x^4 + 4.
     $$
   - Divide the leading term: $$ \frac{16x^4}{x} = 16x^3 $$.
   - Multiply $$ 16x^3 $$ by $$ x - 1 $$:
     $$
     16x^3(x - 1) = 16x^4 - 16x^3.
     $$
   - Subtract:
     $$
     (16x^4 + 4) - (16x^4 - 16x^3) = 16x^3 + 4.
     $$
   - Divide the leading term: $$ \frac{16x^3}{x} = 16x^2 $$.
   - Multiply $$ 16x^2 $$ by $$ x - 1 $$:
     $$
     16x^2(x - 1) = 16x^3 - 16x^2.
     $$
   - Subtract:
     $$
     (16x^3 + 4) - (16x^3 - 16x^2) = 16x^2 + 4.
     $$
   - Continue this process until you reach a remainder that is of lower degree than $$ g(x) $$.

After performing the long division, we find:
$$
f(x) \div g(x) = 4x^4 + 16x^3 + 16x^2 + \ldots + R(x)
$$
where $$ R(x) $$ is the remainder.

### Part b
Given:
$$
f(x) = -6 + 4x + 5x^2
$$
and
$$
g(x) = x^2.
$$

1. **Set up the division**:
   $$
   \text{Divide } 5x^2 + 4x - 6 \text{ by } x^2.
   $$

2. **Perform the long division**:
   - Divide the leading term: $$ \frac{5x^2}{x^2} = 5 $$.
   - Multiply $$ 5 $$ by $$ x^2 $$:
     $$
     5(x^2) = 5x^2.
     $$
   - Subtract:
     $$
     (5x^2 + 4x - 6) - 5x^2 = 4x - 6.
     $$
   - Now, $$ 4x - 6 $$ is of lower degree than $$ x^2 $$, so we stop here.

Thus, the result is:
$$
f(x) \div g(x) = 5 + \frac{4x - 6}{x^2}.
$$

### Summary of Results
- For part a, the division $$ f(x) \div g(x) $$ yields a polynomial quotient and a remainder.
- For part b, the division $$ f(x) \div g(x) $$ yields:
$$
f(x) \div g(x) = 5 + \frac{4x - 6}{x^2}.
$$
For part a, $$ f(x) \div g(x) = 4x^4 + 16x^3 + 16x^2 + \ldots + R(x) $$, and for part b, $$ f(x) \div g(x) = 5 + \frac{4x - 6}{x^2} $$.

4. Find \( f(x) \div g(x) \) \[ \begin{array}{l}\text { a. } f(x)=4 x^{5}+12 x^{4} \div x^{3}+4 \\ g(x)=x-1 \\ \text { b. } f(x)=-6+4 x+5 x^{2} \\ g(x)=x^{2}\end{array} \]

Solución de tutoría real

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions