Answer :
To find the Highest Common Factor (H.C.F.) of the given algebraic expressions [tex]\( x^3 + 7x^2 + 12x \)[/tex], [tex]\( x^2 - 84 \)[/tex], and [tex]\( x^2 - 16 \)[/tex], follow these steps:
1. Identify each expression:
- [tex]\( f(x) = x^3 + 7x^2 + 12x \)[/tex]
- [tex]\( g(x) = x^2 - 84 \)[/tex]
- [tex]\( h(x) = x^2 - 16 \)[/tex]
2. Understand what H.C.F. means for algebraic expressions:
- The H.C.F. of algebraic expressions is the largest expression (in terms of degree and coefficients) that divides each of the given expressions without leaving a remainder.
3. Check the divisibility of common factors:
- We need to find a polynomial that divides all the three expressions. The most straightforward candidates for common factors are polynomials with lower degrees that can potentially divide higher degree polynomials.
4. Calculate the H.C.F. analytically or through tools (implied):
- In this context, the result indicates a simple result related to the argument within the polynomial expression.
5. Conclusion:
- It turns out that after the computations or considerations, the expressions do not share any common factors other than a constant. Therefore, the H.C.F. for these expressions is simply [tex]\( 1 \)[/tex].
This implies that the given algebraic expressions are prime with respect to each other, meaning there is no higher degree polynomial other than [tex]\( 1 \)[/tex] that divides all of them.
1. Identify each expression:
- [tex]\( f(x) = x^3 + 7x^2 + 12x \)[/tex]
- [tex]\( g(x) = x^2 - 84 \)[/tex]
- [tex]\( h(x) = x^2 - 16 \)[/tex]
2. Understand what H.C.F. means for algebraic expressions:
- The H.C.F. of algebraic expressions is the largest expression (in terms of degree and coefficients) that divides each of the given expressions without leaving a remainder.
3. Check the divisibility of common factors:
- We need to find a polynomial that divides all the three expressions. The most straightforward candidates for common factors are polynomials with lower degrees that can potentially divide higher degree polynomials.
4. Calculate the H.C.F. analytically or through tools (implied):
- In this context, the result indicates a simple result related to the argument within the polynomial expression.
5. Conclusion:
- It turns out that after the computations or considerations, the expressions do not share any common factors other than a constant. Therefore, the H.C.F. for these expressions is simply [tex]\( 1 \)[/tex].
This implies that the given algebraic expressions are prime with respect to each other, meaning there is no higher degree polynomial other than [tex]\( 1 \)[/tex] that divides all of them.
