To find the Highest Common Factor (HCF) of the given expressions, we will follow a methodical approach that involves factorization and identification of common factors. The expressions given are:
[tex]\[ a) \quad 2x^2(x+2)(x-2) \][/tex]
[tex]\[ b) \quad 4x(x+2)(x+3) \][/tex]
Let's take each expression and simplify them by noting their factors individually.
1. Expression [tex]\(2x^2(x+2)(x-2)\)[/tex]:
Here, the expression can be rewritten as:
[tex]\[ 2x^2 \cdot (x+2) \cdot (x-2) \][/tex]
2. Expression [tex]\(4x(x+2)(x+3)\)[/tex]:
Similarly, the second expression can be written as:
[tex]\[ 4x \cdot (x+2) \cdot (x+3) \][/tex]
Next, we need to identify the common factors between these two expressions. Let's list out the factors explicitly:
- The first expression [tex]\(2x^2(x+2)(x-2)\)[/tex] has the factors: [tex]\(2\)[/tex], [tex]\(x^2\)[/tex], [tex]\((x+2)\)[/tex], and [tex]\((x-2)\)[/tex].
- The second expression [tex]\(4x(x+2)(x+3)\)[/tex] has the factors: [tex]\(4\)[/tex], [tex]\(x\)[/tex], [tex]\((x+2)\)[/tex], and [tex]\((x+3)\)[/tex].
Now, let's identify the common factors from both lists:
- Common numeric factor: The numeric factors in the expressions are [tex]\(2\)[/tex] and [tex]\(4\)[/tex]. The HCF of these numbers is [tex]\(2\)[/tex].
- Common [tex]\(x\)[/tex] factor: The first expression has [tex]\(x^2\)[/tex] (which is [tex]\(x \cdot x\)[/tex]), and the second expression has [tex]\(x\)[/tex]. The common factor here would be [tex]\(x\)[/tex].
- Common polynomial factor: Both expressions have [tex]\((x+2)\)[/tex] as a factor.
Putting these common factors together, we get the HCF of the given expressions:
[tex]\[ HCF = 2 \cdot x \cdot (x+2) \][/tex]
Simplifying this product, we get:
[tex]\[ HCF = 2x(x+2) \][/tex]
Therefore, the Highest Common Factor (HCF) of the expressions [tex]\(2x^2(x+2)(x-2)\)[/tex] and [tex]\(4x(x+2)(x+3)\)[/tex] is:
[tex]\[ \boxed{2x^2 + 4x} \][/tex]
This is the simplified form of the HCF, clearly showing the steps and the common factors involved.
