Certainly! Let's solve this problem step-by-step.
### Part 1: Finding the Greatest Algebraic Expression that Divides \(\(4x^3\)\) and \(\(6x\)\)
First, let's factor each expression:
- \(\(4x^3\)\): This can be factored as \(\(4 \cdot x^3 = 2^2 \cdot x^3\)\).
- \(\(6x\)\): This can be factored as \(\(6 \cdot x = 2 \cdot 3 \cdot x\)\).
To find the greatest common factor (GCF), identify the lowest powers of all common factors:
- The common numerical factor is \(\(2\)\) (since \(\(2\)\) is common and the lowest power is \(\(2^1\)\)).
- The common variable factor is \(\(x\)\) (since \(\(x\)\) is common and the lowest power is \(\(x^1\)\)).
So, the GCF of \(\(4x^3\)\) and \(\(6x\)\) is:
\(\[ 2 \cdot x = 2x \]\)
### Part 2: Finding the Greatest Algebraic Expression that Exactly Divides \(\((x^2 + 5x + 6)\)\) and \(\((x^2 + x - 6)\)\)
First, let's factor each polynomial:
1. Factorizing \(\(x^2 + 5x + 6\)\):
- Find two numbers that multiply to \(\(6\)\) and add up to \(\(5\)\). These numbers are \(\(2\)\) and \(\(3\)\).
- So, \(\(x^2 + 5x + 6 = (x + 2)(x + 3)\)\).
2. Factorizing \(\(x^2 + x - 6\)\):
- Find two numbers that multiply to \(\(-6\)\) and add up to \(\(1\)\). These numbers are \(\(3\)\) and \(\(-2\)\).
- So, \(\(x^2 + x - 6 = (x + 3)(x - 2)\)\).
Now, identify the common factors in both factorizations:
- \(\((x + 2)(x + 3)\)\)
- \(\((x + 3)(x - 2)\)\)
The common factor is:
\(\[ x + 3 \]\)
### Summary of Results:
1. The greatest algebraic expression that divides \(\(4x^3\)\) and \(\(6x\)\) without leaving a remainder is \(\(2x\)\).
2. The greatest algebraic expression that exactly divides \(\((x^2 + 5x + 6)\)\) and \(\((x^2 + x - 6)\)\) is \(\(x + 3\)\).
