College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Factor completely:

[tex]24x^5 - 36x^4 + 60x^3[/tex]

A. [tex]12x^3(2x^2 - 3x + 5)[/tex]
B. [tex]12x^2(2x^3 - 3x^2 + 5x)[/tex]
C. [tex]6x^2(4x^3 - 6x^2 + 10x)[/tex]
D. [tex]6x^3(4x^2 - 6x + 10)[/tex]

Answer :

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].