Let's factorize the given trinomial step-by-step. The trinomial we need to factorize is:
[tex]\[ 3x^3 - 24x^2 + 45x \][/tex]
Step 1: Factor out the greatest common factor (GCF) from all the terms.
The GCF of the terms [tex]\(3x^3\)[/tex], [tex]\(-24x^2\)[/tex], and [tex]\(45x\)[/tex] is [tex]\(3x\)[/tex]. So we factor out [tex]\(3x\)[/tex] from each term:
[tex]\[ 3x^3 - 24x^2 + 45x = 3x ( x^2 - 8x + 15 ) \][/tex]
Step 2: Factor the quadratic expression inside the parentheses.
The quadratic expression we need to factor is:
[tex]\[ x^2 - 8x + 15 \][/tex]
To factor this quadratic, we look for two numbers that multiply to the constant term (15) and add up to the coefficient of the linear term (-8).
The two numbers that work are -3 and -5, because:
[tex]\[ -3 \times -5 = 15 \][/tex]
[tex]\[ -3 + (-5) = -8 \][/tex]
Therefore, we can write:
[tex]\[ x^2 - 8x + 15 = (x - 3)(x - 5) \][/tex]
Step 3: Substitute the factored quadratic expression back into the expression from Step 1.
So, we get:
[tex]\[ 3x ( x^2 - 8x + 15 ) = 3x ( x - 3 )( x - 5 ) \][/tex]
So, the factorized form of the trinomial [tex]\( 3x^3 - 24x^2 + 45x \)[/tex] is:
[tex]\[ 3x ( x - 3 ) ( x - 5 ) \][/tex]
Thus, the complete factorization of the trinomial is:
[tex]\[ \boxed{3x ( x - 3 ) ( x - 5 )} \][/tex]
