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