BIG IDEAS MATH \#3 i Find the quiotient. Write your answer in factored form. $$ \frac{16 x^{2}-8 x+1}{x^{3}-7 x^{2}+12 x} \div \frac{20 x^{2}-5 x}{15 x^{3}}=\square, x
eq \square, x
eq \square 0 $$ Previous

Question

BIG IDEAS MATH \#3 i Find the quiotient. Write your answer in factored form. $$ \frac{16 x^{2}-8 x+1}{x^{3}-7 x^{2}+12 x} \div \frac{20 x^{2}-5 x}{15 x^{3}}=\square, x 
eq \square, x 
eq \square 0 $$ Previous

UpStudy AI · Accepted Answer

To solve the problem, we need to find the quotient of the two rational expressions given. The expression is:

$$
\frac{16 x^{2}-8 x+1}{x^{3}-7 x^{2}+12 x} \div \frac{20 x^{2}-5 x}{15 x^{3}}
$$

### Step 1: Rewrite the Division as Multiplication

We can rewrite the division of fractions as multiplication by the reciprocal:

$$
\frac{16 x^{2}-8 x+1}{x^{3}-7 x^{2}+12 x} \times \frac{15 x^{3}}{20 x^{2}-5 x}
$$

### Step 2: Factor Each Expression

Now, we will factor each part of the expression.

1. **Numerator of the first fraction**: $$16 x^{2}-8 x+1$$

This is a quadratic expression. We can factor it as follows:

$$
   16 x^{2}-8 x+1 = (4x-1)^{2}
   $$

2. **Denominator of the first fraction**: $$x^{3}-7 x^{2}+12 x$$

We can factor out $$x$$:

$$
   x(x^{2}-7x+12)
   $$

Now, we can factor the quadratic $$x^{2}-7x+12$$:

$$
   x^{2}-7x+12 = (x-3)(x-4)
   $$

So, the complete factorization is:

$$
   x(x-3)(x-4)
   $$

3. **Numerator of the second fraction**: $$20 x^{2}-5 x$$

We can factor out $$5x$$:

$$
   5x(4x-1)
   $$

4. **Denominator of the second fraction**: $$15 x^{3}$$

This is already factored as $$15x^{3}$$.

### Step 3: Substitute the Factored Forms

Now we substitute the factored forms back into the expression:

$$
\frac{(4x-1)^{2}}{x(x-3)(x-4)} \times \frac{15x^{3}}{5x(4x-1)}
$$

### Step 4: Simplify the Expression

Now we can simplify the expression:

1. Cancel $$5x$$ in the numerator and denominator:

$$
   \frac{(4x-1)^{2}}{(x-3)(x-4)} \times \frac{3x^{2}}{(4x-1)}
   $$

2. Cancel one $$4x-1$$:

$$
   \frac{(4x-1)}{(x-3)(x-4)} \times 3x^{2}
   $$

3. Combine the fractions:

$$
   \frac{3x^{2}(4x-1)}{(x-3)(x-4)}
   $$

### Final Answer

The final answer in factored form is:

$$
\frac{3x^{2}(4x-1)}{(x-3)(x-4)}
$$

### Restrictions

The restrictions based on the original denominators are:

1. $$x \neq 0$$
2. $$x \neq 3$$
3. $$x \neq 4$$

Thus, the final answer is:

$$
\frac{3x^{2}(4x-1)}{(x-3)(x-4)}, \quad x \neq 0, x \neq 3, x \neq 4
$$
The quotient in factored form is $$ \frac{3x^{2}(4x-1)}{(x-3)(x-4)} $$, with restrictions $$ x \neq 0, 3, 4 $$.

BIG IDEAS MATH \#3 i Find the quiotient. Write your answer in factored form. \( \frac{16 x^{2}-8 x+1}{x^{3}-7 x^{2}+12 x} \div \frac{20 x^{2}-5 x}{15 x^{3}}=\square, x \neq \square, x \neq \square 0 \) Previous

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Algebra Questions