Arrange the steps of the following factor by grouping problem: $$ 18 x^{4}-9 x^{3}-8 x^{2}+4 x $$ 1. $$ x\left(18 x^{3}-9 x^{2}-8 x+4\right) $$ 2. $$ x\left(9 x^{2}(2 x-1)-4(2 x-1)\right) $$ 3. $$ x(2 x-1)\left((3 x)^{2}-2^{2}\right) $$ 4. $$ x(2 x-1)(3 x-2)(3 x+2) $$ 5. $$ x\left((2 x-1)\left(9 x^{2}-4\right)\right) $$ Your Score: 2 of 5

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UpStudy AI · Accepted Answer

To factor the expression $$ 18 x^{4}-9 x^{3}-8 x^{2}+4 x $$ by grouping, we can follow these steps:

1. **Factor out the greatest common factor (GCF)** from the entire expression. The GCF here is $$ x $$:
   $$
   x(18 x^{3}-9 x^{2}-8 x+4)
   $$

2. **Group the terms** inside the parentheses:
   $$
   x\left(18 x^{3}-9 x^{2}-8 x+4\right)
   $$

3. **Factor by grouping** the terms $$ 18 x^{3}-9 x^{2} $$ and $$ -8 x+4 $$:
   - From the first group $$ 18 x^{3}-9 x^{2} $$, we can factor out $$ 9 x^{2} $$:
     $$
     9 x^{2}(2 x-1)
     $$
   - From the second group $$ -8 x+4 $$, we can factor out $$ -4 $$:
     $$
     -4(2 x-1)
     $$

4. **Combine the factored groups**:
   $$
   x\left(9 x^{2}(2 x-1)-4(2 x-1)\right)
   $$

5. **Factor out the common binomial factor $$ (2 x-1) $$**:
   $$
   x(2 x-1)\left(9 x^{2}-4\right)
   $$

6. **Recognize that $$ 9 x^{2}-4 $$ is a difference of squares**:
   $$
   9 x^{2}-4 = (3 x)^{2}-2^{2} = (3 x-2)(3 x+2)
   $$

7. **Combine everything**:
   $$
   x(2 x-1)(3 x-2)(3 x+2)
   $$

Now, let's arrange the steps in the correct order:

1. $$ x\left(18 x^{3}-9 x^{2}-8 x+4\right) $$  (Step 1)
2. $$ x\left(9 x^{2}(2 x-1)-4(2 x-1)\right) $$  (Step 4)
3. $$ x(2 x-1)\left((3 x)^{2}-2^{2}\right) $$  (Step 5)
4. $$ x(2 x-1)(3 x-2)(3 x+2) $$  (Step 6)
5. $$ x\left((2 x-1)\left(9 x^{2}-4\right)\right) $$  (Step 3)

So the correct order of steps is:
1. Step 1
2. Step 2
3. Step 5
4. Step 4
5. Step 3

This gives us the final factorization:
$$
x(2 x-1)(3 x-2)(3 x+2)
$$
The correct order of steps is: 1. $$ x\left(18 x^{3}-9 x^{2}-8 x+4\right) $$ 2. $$ x\left(9 x^{2}(2 x-1)-4(2 x-1)\right) $$ 3. $$ x(2 x-1)\left((3 x)^{2}-2^{2}\right) $$ 4. $$ x(2 x-1)(3 x-2)(3 x+2) $$ 5. $$ x\left((2 x-1)\left(9 x^{2}-4\right)\right) $$ Final factorization: $$ x(2 x-1)(3 x-2)(3 x+2) $$

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