Certainly! Let's evaluate and simplify the given expression step-by-step. The expression is:
[tex]\[ 40c^3 + 20bc - 100c^2 \][/tex]
Firstly, identify each of the terms in the expression:
1. [tex]\( 40c^3 \)[/tex] is a cubic term in [tex]\( c \)[/tex].
2. [tex]\( 20bc \)[/tex] is a linear term in both [tex]\( b \)[/tex] and [tex]\( c \)[/tex].
3. [tex]\( -100c^2 \)[/tex] is a quadratic term in [tex]\( c \)[/tex].
To simplify this expression further, we should look for a common factor in each term.
### Step-by-Step Solution:
1. Identify Common Factors:
Observe that each term in the expression [tex]\( 40c^3 + 20bc - 100c^2 \)[/tex] can be factored by identifying the greatest common factor (GCF):
- The GCF for the coefficients [tex]\(40, 20,\)[/tex] and [tex]\(100\)[/tex] is [tex]\(20\)[/tex].
- The common variable factor is [tex]\(c\)[/tex] (since all terms have at least one [tex]\(c\)[/tex] in them).
2. Factor out the Common Elements:
Extract the common factor [tex]\(20c\)[/tex] from the expression. This will make it easier to simplify further:
[tex]\[ 40c^3 + 20bc - 100c^2 = 20c \left( 2c^2 + b - 5c \right) \][/tex]
So, the given expression [tex]\( 40c^3 + 20bc - 100c^2 \)[/tex] can be factored to:
[tex]\[ 20bc + 40c^3 - 100c^2 \][/tex]
By factoring out the common factor, the simplified form of [tex]\( 40c^3 + 20bc - 100c^2 \)[/tex] is:
[tex]\[ \boxed{20c (2c^2 + b - 5c)} \][/tex]
That's the detailed, step-by-step solution for simplifying the given expression.
