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Answer :
To find the highest common factor (HCF) of the given algebraic expressions, we'll consider each expression separately and then determine their common factors.
The expressions are:
1. [tex]\( 12x \)[/tex]
2. [tex]\( x^3 + 44 \)[/tex]
3. [tex]\( x^2 - 16 \)[/tex]
Step-by-step solution:
1. Identify any factors:
- For [tex]\( 12x \)[/tex], the factors are 12 and [tex]\( x \)[/tex]. These can break down further as 2, 2, 3, and [tex]\( x \)[/tex]. There are no common variables to factor out from the remaining terms.
- For [tex]\( x^3 + 44 \)[/tex], it doesn't seem to factor easily into simpler polynomial forms using standard techniques. Checking for factorization based on obvious substitutions (such as looking for roots) doesn't help either since it results in non-integer values.
- For [tex]\( x^2 - 16 \)[/tex], notice that this is a difference of squares. It can be factored as [tex]\( (x - 4)(x + 4) \)[/tex].
2. Find common factors:
- There doesn't seem to be an obvious variable or numerical common factor across all three expressions.
- The factorization of the expressions implies no shared algebraic term between them.
3. Conclusion:
- The lack of a common number or polynomial expression across the three expressions indicates that their only greatest common factor is 1. This means there are no other common factors aside from 1 that can divide all the terms simultaneously.
Thus, the highest common factor (HCF) of [tex]\( 12x \)[/tex], [tex]\( x^3 + 44 \)[/tex], and [tex]\( x^2 - 16 \)[/tex] is 1.
The expressions are:
1. [tex]\( 12x \)[/tex]
2. [tex]\( x^3 + 44 \)[/tex]
3. [tex]\( x^2 - 16 \)[/tex]
Step-by-step solution:
1. Identify any factors:
- For [tex]\( 12x \)[/tex], the factors are 12 and [tex]\( x \)[/tex]. These can break down further as 2, 2, 3, and [tex]\( x \)[/tex]. There are no common variables to factor out from the remaining terms.
- For [tex]\( x^3 + 44 \)[/tex], it doesn't seem to factor easily into simpler polynomial forms using standard techniques. Checking for factorization based on obvious substitutions (such as looking for roots) doesn't help either since it results in non-integer values.
- For [tex]\( x^2 - 16 \)[/tex], notice that this is a difference of squares. It can be factored as [tex]\( (x - 4)(x + 4) \)[/tex].
2. Find common factors:
- There doesn't seem to be an obvious variable or numerical common factor across all three expressions.
- The factorization of the expressions implies no shared algebraic term between them.
3. Conclusion:
- The lack of a common number or polynomial expression across the three expressions indicates that their only greatest common factor is 1. This means there are no other common factors aside from 1 that can divide all the terms simultaneously.
Thus, the highest common factor (HCF) of [tex]\( 12x \)[/tex], [tex]\( x^3 + 44 \)[/tex], and [tex]\( x^2 - 16 \)[/tex] is 1.
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