Alright, let's tackle the problem of factoring the polynomial expression [tex]\( 24 x^5 - 36 x^4 + 60 x^3 \)[/tex].
To factor this expression completely, we'll follow these steps:
1. Find the greatest common factor (GCF) of the coefficients and the powers of [tex]\( x \)[/tex].
2. Factor out the GCF from the given polynomial.
3. Express the remaining polynomial in its simplest factored form.
Step 1: Identify the Greatest Common Factor (GCF)
Look at the coefficients of [tex]\( x \)[/tex] in the polynomial [tex]\( 24 x^5 - 36 x^4 + 60 x^3 \)[/tex].
The GCF of 24, 36, and 60 is 12.
For the variables, the smallest power of [tex]\( x \)[/tex] is [tex]\( x^3 \)[/tex], so the GCF for the entire expression is [tex]\( 12 x^3 \)[/tex].
Step 2: Factor out the GCF
Factor [tex]\( 12 x^3 \)[/tex] out of each term in the polynomial:
[tex]\[
24 x^5 - 36 x^4 + 60 x^3 = 12 x^3 \left(\frac{24 x^5}{12 x^3}\right) - 12 x^3 \left(\frac{36 x^4}{12 x^3}\right) + 12 x^3 \left(\frac{60 x^3}{12 x^3}\right)
\][/tex]
This simplifies to:
[tex]\[
24 x^5 - 36 x^4 + 60 x^3 = 12 x^3 (2 x^2) - 12 x^3 (3 x) + 12 x^3 (5)
\][/tex]
[tex]\[
= 12 x^3 (2 x^2 - 3 x + 5)
\][/tex]
Step 3: Verify the remaining polynomial is in simplest form
The polynomial inside the parentheses, [tex]\( 2 x^2 - 3 x + 5 \)[/tex], cannot be factored further using real numbers because the discriminant [tex]\((b^2 - 4ac)\)[/tex] [tex]\(= (-3)^2 - 425 = 9 - 40 = -31\)[/tex], which is negative. Hence it does not have real roots.
Therefore, the completely factored form of the polynomial [tex]\( 24 x^5 - 36 x^4 + 60 x^3 \)[/tex] is:
[tex]\[
12 x^3 (2 x^2 - 3 x + 5)
\][/tex]
Which matches option (A):
[tex]\[
12 x^3\left(2 x^2-3 x+5\right)
\][/tex]
So, the correct answer is:
(A) [tex]\( 12 x^3\left(2 x^2 - 3 x + 5\right) \)[/tex].
