Question

unit factoring
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factor out the greatest common factor (gcf): ( 45d^3 - 18d^2 )
( 9d(5d^2 - 2d) )
( 9d^2 cdot 5d - 9d^2 cdot 2 )
( 9d^2(5d - 2) )
( 3d(15d^2 - 6d) )

Explanation:

Step1: Find GCF of coefficients and variables

For \(45d^3 - 18d^2\), GCF of 45 and 18 is 9. GCF of \(d^3\) and \(d^2\) is \(d^2\). So GCF is \(9d^2\)? Wait, no, wait: 45 and 18: factors of 45: 1,3,5,9,15,45; factors of 18:1,2,3,6,9,18. GCF is 9. For variables: \(d^3 = d^2 \cdot d\), \(d^2 = d^2 \cdot 1\), so GCF of variables is \(d^2\)? Wait no, wait the terms are \(45d^3\) and \(-18d^2\). Wait, maybe I miscalculated. Wait 45d³ -18d². Let's factor each term:

45d³ = 9 * 5 * d² * d

-18d² = 9 * (-2) * d²

So GCF is 9d²? Wait no, wait the options have 9d²(5d - 2)? Wait wait, let's check the options. Wait the options are:

1. 9d(5d² - 2d)

2. 9d²·5d - 9d²·2

3. 9d²(5d - 2)

4. 3d(15d² - 6d)

Wait let's factor 45d³ -18d².

First, find GCF of 45 and 18: 9.

GCF of d³ and d²: d².

So GCF is 9d².

Now factor out 9d² from 45d³ -18d²:

45d³ -18d² = 9d²(5d) - 9d²(2) = 9d²(5d - 2).

Wait but let's check the options. Option 3 is 9d²·5d - 9d²·2? No, option 3 is 9d²(5d - 2)? Wait the third option is "9d²·5d - 9d²·2"? Wait no, the user's image: let's re-express the options:

Option 1: 9d(5d² - 2d)

Option 2: 9d²·5d - 9d²·2 (this is just expanding, not factoring out, it's the original terms split)

Option 3: 9d²(5d - 2)

Option 4: 3d(15d² - 6d)

Wait let's factor 45d³ -18d²:

Take GCF 9d²:

45d³ -18d² = 9d²(5d) - 9d²(2) = 9d²(5d - 2). Which is option 3? Wait no, the third option in the image: "9d²(5d - 2)"? Wait the user's image: the third option is "9d²(5d - 2)"? Wait the second option is "9d²·5d - 9d²·2", third is "9d²(5d - 2)"? Wait no, let's check the original problem again.

Wait the problem is to factor out GCF from 45d³ -18d².

Let's compute GCF of 45 and 18: 9.

GCF of d³ and d²: d².

So GCF is 9d².

Now, 45d³ ÷ 9d² = 5d.

-18d² ÷ 9d² = -2.

So factoring out 9d² gives 9d²(5d - 2). Which is the third option? Wait the options:

First option: 9d(5d² - 2d) → 9d*5d²=45d³, 9d*(-2d)=-18d². Wait that's also correct? Wait no, 9d(5d² - 2d) = 45d³ - 18d². But is that factoring out the GCF? Wait GCF of 45d³ and -18d²: let's check GCF of 45 and 18 is 9, GCF of d³ and d² is d²? Wait no, 45d³ and -18d²: GCF is 9d²? Wait 45d³ = 9d²*5d, -18d²=9d²*(-2). So GCF is 9d². But 9d(5d² - 2d) = 9d*5d² -9d*2d =45d³ -18d². But here, the GCF factored out is 9d, not 9d². So which is the GCF?

Wait GCF is the greatest common factor. Let's list the factors of 45d³ and -18d²:

Prime factors of 45: 3²*5

Prime factors of 18: 2*3²

So GCF of coefficients: 3²=9.

Prime factors of d³: d*d*d

Prime factors of d²: d*d

So GCF of variables: d*d = d².

Thus, GCF is 9d². So factoring out 9d² gives 9d²(5d - 2), which is option 3 (if option 3 is 9d²(5d - 2)). Wait the user's image: the third option is "9d²(5d - 2)"? Wait the second option is "9d²·5d - 9d²·2" (which is the distributive property expanded, not factored), the third option is "9d²(5d - 2)"? Wait no, looking at the image:

The options are:

1. 9d(5d² - 2d)

2. 9d²·5d - 9d²·2

3. 9d²(5d - 2)

4. 3d(15d² - 6d)

Wait let's check each:

Option 1: 9d(5d² - 2d) = 45d³ - 18d². Correct, but is this factoring out the GCF? GCF here would be 9d, but the actual GCF is 9d². Wait no, 45d³ = 9d * 5d², -18d² = 9d * (-2d). So GCF is 9d? But earlier we thought GCF is 9d². Wait, maybe I made a mistake. Let's check the exponents:

45d³: degree 3

-18d²: degree 2

The GCF's degree is the minimum of the degrees, which is 2? No, wait GCF of variables: for d³ and d², the GCF is d² (since d² divides both d³ and d²). Wait d³ ÷ d² = d, d² ÷ d² = 1. So GCF of variables is d². So GCF is 9d². Then factoring out 9d² gives 9d²(5d - 2), which is option 3. Let's check: 9d²*5d = 45d³, 9d²*(-2) = -18d². Yes, that's correct. So the correct answer is the third option: 9d²(5d - 2), which is option 3 (if the third option is written as 9d²(5d - 2)). Wait the user's image: the third option is "9d²(5d - 2)"? Wait the second option is "9d²·5d - 9d²·2" (which is 9d²(5d - 2) expanded), the third option is "9d²(5d - 2)". Wait no, looking at the image:

The options are:

1. 9d(5d² - 2d)

2. 9d²·5d - 9d²·2

3. 9d²(5d - 2)

4. 3d(15d² - 6d)

So option 3 is factoring out 9d², which is the GCF. So that's the correct answer.

Explanation:

Step1: Factor each term

Factor \(45d^3 = 9d^2 \cdot 5d\) and \(-18d^2 = 9d^2 \cdot (-2)\).

Step2: Identify GCF

The greatest common factor (GCF) of \(45d^3\) and \(-18d^2\) is \(9d^2\).

Step3: Factor out GCF

Factor out \(9d^2\) from \(45d^3 - 18d^2\):
\[ 45d^3 - 18d^2 = 9d^2(5d) + 9d^2(-2) = 9d^2(5d - 2) \]

Step1: Factor each term

Factor \(45d^3 = 9d^2 \cdot 5d\) and \(-18d^2 = 9d^2 \cdot (-2)\).

Step2: Identify GCF

The greatest common factor (GCF) of \(45d^3\) and \(-18d^2\) is \(9d^2\).

Step3: Factor out GCF

Factor out \(9d^2\) from \(45d^3 - 18d^2\):

$$ 45d^3 - 18d^2 = 9d^2(5d) + 9d^2(-2) = 9d^2(5d - 2) $$

Answer:

9d²(5d - 2) (the third option, e.g., if options are labeled as 1,2,3,4, the correct one is the third option with text "9d²(5d - 2)")