Sure, let's take each part of the question in turn and find the Highest Common Factor (HCF) of the given polynomials.
### (a) [tex]\( a^5 + b^3, a^3 - a^2 b + ab^2, a^4 + a^2 b^2 + b^4 \)[/tex]
To find the HCF of these polynomials, examine common factors. Polynomials involving different powers of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] without common terms suggest their HCF is 1.
Answer: [tex]\( \text{HCF} = 1 \)[/tex]
### (b) [tex]\( x^2 + 2x^2 + 2x + 1, x^3 - 1, x^4 + x^2 + 1 \)[/tex]
Combine like terms in the first polynomial: [tex]\( 3x^2 + 2x + 1 \)[/tex]. For three polynomials with different powers and complex terms, finding common factors yields 1.
Answer: [tex]\( \text{HCF} = 1 \)[/tex]
### (c) [tex]\( x^3 - 4x, 4x^3 - 10x^2 + 4x, 3x^4 - 8x^3 + 4x^2 \)[/tex]
These polynomials share a common factor based on the variable [tex]\( x \)[/tex]:
- [tex]\( x^3 - 4x \)[/tex] factors as [tex]\( x(x^2 - 4) \)[/tex].
- [tex]\( 4x^3 - 10x^2 + 4x \)[/tex] factors as [tex]\( 2x(2x^2 - 5x + 2) \)[/tex].
- [tex]\( 3x^4 - 8x^3 + 4x^2 \)[/tex] factors as [tex]\( x^2 (3x^2 - 8x + 4) \)[/tex].
The common factor among these polynomials is [tex]\( x \cdot x = x^2 - 2x \)[/tex].
Answer: [tex]\( \text{HCF} = x^2 - 2x \)[/tex]
### (d) [tex]\( y^2 + 2y - 8, y^2 - 5y + 6, y^2 + 5y - 14 \)[/tex]
Simplify the equations to find common linear factors:
- [tex]\( y^2 + 2y - 8 \)[/tex] factors as [tex]\( (y + 4)(y - 2) \)[/tex].
- [tex]\( y^2 - 5y + 6 \)[/tex] factors as [tex]\( (y - 2)(y - 3) \)[/tex].
- [tex]\( y^2 + 5y - 14 \)[/tex] factors as [tex]\( (y + 7)(y - 2) \)[/tex].
The common factor is [tex]\( y - 2 \)[/tex].
Answer: [tex]\( \text{HCF} = y - 2 \)[/tex]
### (e) [tex]\( x^2 + 2x + 1, x^2 + 5x + 6, 2x^2 - 5x + 2 \)[/tex]
Examining possible common factors:
- [tex]\( x^2 + 2x + 1 \)[/tex] factors as [tex]\( (x + 1)^2 \)[/tex].
- [tex]\( x^2 + 5x + 6 \)[/tex] factors as [tex]\( (x + 2)(x + 3) \)[/tex].
- [tex]\( 2x^2 - 5x + 2 \)[/tex] factors as [tex]\( (2x - 1)(x - 2) \)[/tex].
These polynomials do not share common factors other than 1.
Answer: [tex]\( \text{HCF} = 1 \)[/tex]
### (f) [tex]\( x^2 + 2x - 8, x^2 - 2x - 24, x^2 + 5x + 4 \)[/tex]
Examining possible common factors:
- [tex]\( x^2 + 2x - 8 \)[/tex] factors as [tex]\( (x + 4)(x - 2) \)[/tex].
- [tex]\( x^2 - 2x - 24 \)[/tex] factors as [tex]\( (x - 6)(x + 4) \)[/tex].
- [tex]\( x^2 + 5x + 4 \)[/tex] factors as [tex]\( (x + 1)(x + 4) \)[/tex].
The common factor is [tex]\( x + 4 \)[/tex].
Answer: [tex]\( \text{HCF} = x + 4 \)[/tex]
