24x^4 - 28x^3 + 8 x^2 can be factored as cx^2(dx-2)(ex-1), what is value of d-c

Given the expression \( 24x^4 - 28x^3 + 8x^2 \), we are asked to factor it in the form \( cx^2(dx - 2)(ex - 1) \) and find the value of \( d - c \).

### Step 1: Factor out the common factor
The given expression is:
\[
24x^4 - 28x^3 + 8x^2
\]
First, we factor out the greatest common factor from all the terms, which is \( 4x^2 \):
\[
4x^2(6x^2 - 7x + 2)
\]

### Step 2: Factor the quadratic expression
Now, we need to factor the quadratic \( 6x^2 - 7x + 2 \). We seek two numbers that multiply to \( 6 \times 2 = 12 \) and add to \( -7 \). The numbers are \( -3 \) and \( -4 \).

So, we rewrite the middle term \( -7x \) as \( -3x - 4x \):
\[
6x^2 - 3x - 4x + 2
\]
Now, factor by grouping:
\[
3x(2x - 1) - 2(2x - 1)
\]
This gives us:
\[
(3x - 2)(2x - 1)
\]

### Step 3: Substitute back into the factored form
The full factorization of the original expression is:
\[
4x^2(3x - 2)(2x - 1)
\]

### Step 4: Identify \( c \), \( d \), and \( e \)
Comparing this with the form \( cx^2(dx - 2)(ex - 1) \), we see:
\[
c = 4, \quad d = 3, \quad e = 2
\]

### Step 5: Calculate \( d - c \)
\[
d - c = 3 - 4 = -1
\]

Thus, the value of \( d - c \) is \( \boxed{-1} \).

Do you want more details or have any questions? Here are some related questions you might find interesting:

1. How do you factor quadratic expressions using the grouping method?
2. What are some methods for factoring polynomials of higher degrees?
3. How does identifying the greatest common factor simplify polynomial factoring?
4. What are the applications of factoring in solving polynomial equations?
5. Can you provide an example of factoring a cubic polynomial?

**Tip:** Always start by factoring out the greatest common factor (GCF) before attempting to factor a polynomial completely.

Learn how to factor the polynomial 24x^4 - 28x^3 + 8x^2 into the form cx^2(dx - 2)(ex - 1) using quadratic factoring methods. This detailed solution covers factoring by grouping and understanding polynomial terms. Suitable for high school students.

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