Common Factors and Multiples
Excerpts
Common Factors and Multiples
Resolve each expression into its simplest factors. Find the product of all the common factors, taking each factor the least number of times it occurs in any of the given expressions.
Common Factors and Multiples
We cannot apply the terms *greatest* and *least* to an algebraic expression in which particular values have not been given to the letters contained in the expression. Thus $a$ is *greater* than $a^{2}$, if $a$ stands for $\frac{1}{4}$.
Common Factors and Multiples
The L. C. M. must evidently contain each factor the greatest number of times that it occurs in any expression.
Common Factors and Multiples
The *highest common factor* in Algebra corresponds to the *greatest common measure*, or *greatest common divisor* in Arithmetic.
Common Factors and Multiples
The **highest common factor** of two or more integral and rational *expressions* is an integral and rational expression of highest degree that will divide each of them without a remainder.
Equations
Problems
Exercise 40
Exercise 40, problem 1, p. 86
$330$ and $546$.
Printed answer:- $6$.
verified: the printed answer passed a computed check
How it was checked
hcf: passes6
Exercise 40, problem 10, p. 86
$(x + y)^{2}$ and $x^{2} - y^{2}$.
Printed answer:- $x + y$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx + y
Exercise 40, problem 11, p. 86
$a^{3} + a^{2}x$ and $a^{2} - x^{2}$.
Printed answer:- $a + x$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesa + x
Exercise 40, problem 12, p. 86
$a^{2} - 4b^{2}$ and $a^{2} + 2ab$.
Printed answer:- $a + 2b$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesa + 2*b
Exercise 40, problem 13, p. 86
$x^{2} - 1$ and $x^{2} + 2x - 3$.
Printed answer:- $x - 1$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx - 1
Exercise 40, problem 14, p. 86
$x^{2} + 5x + 6$ and $x^{2} + 4x + 3$.
Printed answer:- $x + 3$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx + 3
Exercise 40, problem 15, p. 86
$x^{2} - 9x + 18$ and $x^{2} - 10x + 24$.
Printed answer:- $x - 6$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx - 6
Exercise 40, problem 16, p. 86
$x^{3} + 1$ and $x^{2} - x + 1$.
Printed answer:- $x^{2} - x + 1$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx**2 - x + 1
Exercise 40, problem 17, p. 86
$x^{2} - 3x + 2$ and $x^{2} - 4x + 3$.
Printed answer:- $x - 1$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx - 1
Exercise 40, problem 18, p. 86
$x^{2} - 3xy + 2y^{2}$ and $x^{2} - 2xy + y^{2}$.
Printed answer:- $x - y$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx - y
Exercise 40, problem 19, p. 86
$x^{2} - 4x - 5$ and $x^{2} - 25$.
Printed answer:- $x - 5$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx - 5
Exercise 40, problem 2, p. 86
$20x^{3}$ and $15x^{4}$.
Printed answer:- $5x^{3}$.
verified: the printed answer passed a computed check
How it was checked
hcf: passes5*x**3
Exercise 40, problem 20, p. 86
$(a - b)^{2} - c^{2}$ and $ab - b^{2} - bc$.
Printed answer:- $a - b - c$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesa - b - c
Exercise 40, problem 21, p. 86
$x^{2} + xy -2y^{2}$ and $x^{2} + 5xy + 6y^{2}$.
Printed answer:- $x + 2y$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx + 2*y
Exercise 40, problem 22, p. 86
$x^{2} + 7xy + 12y^{2}$ and $x^{2} + 3xy - 4y^{2}$.
Printed answer:- $x + 4y$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx + 4*y
Exercise 40, problem 23, p. 86
$x^{3} - 8y^{3}$ and $x^{2} + 2xy + 4y^{2}$.
Printed answer:- $x^{2} + 2xy + 4y^{2}$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx**2 + 2*x*y + 4*y**2
Exercise 40, problem 24, p. 86
$x^{3} - 2x^{2} - x + 2$ and $x^{2} - 4x + 4$.
Printed answer:- $x - 2$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx - 2
Exercise 40, problem 25, p. 86
$1 - 5a + 6a^{2}$ and $1 - 7a + 12a^{2}$.
Printed answer:- $1 - 3a$.
verified: the printed answer passed a computed check
How it was checked
hcf: passes1 - 3*a
Exercise 40, problem 26, p. 86
$x^{2} - 8xy + 7y^{2}$ and $x^{2} - 3xy - 28y^{2}$.
Printed answer:- $x - 7y$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx - 7*y
Exercise 40, problem 27, p. 86
$8a^{3} + b^{3}$ and $4a^{2} + 4ab + b^{2}$.
Printed answer:- $2a + b$.
verified: the printed answer passed a computed check
How it was checked
hcf: passes2*a + b
Exercise 40, problem 28, p. 86
$x^{2} - (y - z)^{2}$ and $(x + y)^{2} - z^{2}$.
Printed answer:- $x + y - z$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx + y - z
Exercise 40, problem 3, p. 86
$42ax^{2}$ and $60a^{2}x$.
Printed answer:- $6ax$.
verified: the printed answer passed a computed check
How it was checked
hcf: passes6*a*x
Exercise 40, problem 4, p. 86
$35a^{2}b^{2}$ and $49ab^{3}$.
Printed answer:- $7ab^{2}$.
verified: the printed answer passed a computed check
How it was checked
hcf: passes7*a*b**2
Exercise 40, problem 5, p. 86
$28x^{4}$ and $63y^{4}$.
Printed answer:- $7$.
verified: the printed answer passed a computed check
How it was checked
hcf: passes7
Exercise 40, problem 6, p. 86
$54a^{2}b^{2}$ and $56a^{3}b^{3}$.
Printed answer:- $2a^{2}b^{2}$.
verified: the printed answer passed a computed check
How it was checked
hcf: passes2*a**2*b**2
Exercise 40, problem 7, p. 86
$x^{3} + 3x^{2}y$ and $x^{3} + 27y^{3}$.
Printed answer:- $x + 3y$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx + 3*y
Exercise 40, problem 8, p. 86
$x^{2} + 3x$ and $x^{2} - 9$.
Printed answer:- $x + 3$.
verified: the printed answer passed a computed check
How it was checked
hcf: passesx + 3
Exercise 40, problem 9, p. 86
$2ax^{3} + x^{3}$ and $8a^{3} + 1$.
Printed answer:- $2a + 1$.
verified: the printed answer passed a computed check
How it was checked
hcf: passes2*a + 1
Exercise 41
Exercise 41, problem 1, p. 88
$9xy^{3}$ and $6x^{2}y$.
Printed answer:- $18x^{2}y^{3}$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes18*x**2*y**3
Exercise 41, problem 10, p. 88
$x^{2} - 1$ and $x^{2} - x$.
Printed answer:- $x(x + 1)(x - 1)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passesx*(x + 1)*(x - 1)
Exercise 41, problem 11, p. 88
$x^{2} + xy$ and $xy + y^{2}$.
Printed answer:- $xy(x + y)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passesx*y*(x + y)
Exercise 41, problem 12, p. 88
$x^{2} + 2x$ and $(x + 2)^{2}$.
Printed answer:- $x(x + 2)^{2}$.
verified: the printed answer passed a computed check
How it was checked
lcm: passesx*(x + 2)**2
Exercise 41, problem 13, p. 88
$a^{2} + 4a + 4$ and $a^{2} + 5a + 6$.
Printed answer:- $(a + 2)(a + 2)(a + 3)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(a + 2)*(a + 2)*(a + 3)
Exercise 41, problem 14, p. 88
$c^{2} + c - 20$ and $c^{2} - c - 30$.
Printed answer:- $(c - 4)(c + 5)(c - 6)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(c - 4)*(c + 5)*(c - 6)
Exercise 41, problem 15, p. 88
$b^{2} + b - 42$ and $b^{2} - 11b + 30$.
Printed answer:- $(b - 5)(b - 6)(b + 7)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(b - 5)*(b - 6)*(b + 7)
Exercise 41, problem 16, p. 88
$y^{2} - 10y + 24$ and $y^{2} + y -20$.
Printed answer:- $(y - 4)(y + 5)(y - 6)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(y - 4)*(y + 5)*(y - 6)
Exercise 41, problem 17, p. 88
$z^{2} + 2z - 35$ and $z^{2} - 11z + 30$.
Printed answer:- $(z - 5)(z - 6)(z + 7)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(z - 5)*(z - 6)*(z + 7)
Exercise 41, problem 18, p. 88
$x^{2} - 64; x^{3} - 64$; and $x + 8$.
Printed answer:- $(x - 4)(x + 8)(x - 8)(x^{2} + 4x + 16)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(x - 4)*(x + 8)*(x - 8)*(x**2 + 4*x + 16)
Exercise 41, problem 19, p. 88
$a^{2} - b^{2}; (a + b)^{2}$; and $(a - b)^{2}$.
Printed answer:- $(a + b)(a + b)(a - b)(a - b)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(a + b)*(a + b)*(a - b)*(a - b)
Exercise 41, problem 2, p. 88
$3abc^{2}$ and $2a^{2}bc^{3}$.
Printed answer:- $6a^{2}bc^{3}$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes6*a**2*b*c**3
Exercise 41, problem 20, p. 88
$4ab(a + b)^{2}$ and $2a^{2}(a^{2} - b^{2})$.
Printed answer:- $4a^{2}b(a + b)(a + b)(a - b)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes4*a**2*b*(a + b)*(a + b)*(a - b)
Exercise 41, problem 21, p. 88
$y^{2} + 7y + 12; y^{2} + 6y + 8$; and $y^{2} + 5y +6$.
Printed answer:- $(y + 2)(y + 3)(y + 4)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(y + 2)*(y + 3)*(y + 4)
Exercise 41, problem 22, p. 88
$x^{2} - 1; x^{3} + x^{2} + x + 1$; and $x^{3} - x^{2} + x - 1$.
Printed answer:- $(x + 1)(x - 1)(x^{2} + 1)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(x + 1)*(x - 1)*(x**2 + 1)
Exercise 41, problem 23, p. 88
$1 - x^{2}; 1 - x^{3}$; and $1 + x$.
Printed answer:- $(1 + x)(1 - x)(1 + x + x^{2})$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(1 - x)*(1 + x)*(1 + x + x**2)
Exercise 41, problem 24, p. 88
$x^{2} + 2xy + y^{2}; x^{2} - y^{2}$; and $x^{2} - 2xy + y^{2}$.
Printed answer:- $(x + y)(x + y)(x - y)(x - y)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(x + y)*(x + y)*(x - y)*(x - y)
Exercise 41, problem 25, p. 88
$x^{3} - 27; x^{2} + 2x - 15; x^{2} + 5x$.
Printed answer:- $x(x - 3)(x + 5)(x^{2} + 3x + 9)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passesx*(x - 3)*(x + 5)*(x**2 + 3*x + 9)
Exercise 41, problem 26, p. 88
$(a + b)^{2} - c^{2}; (a + b + c)^{2}$; and $a + b - c$.
Printed answer:- $(a + b + c)(a + b + c)(a + b - c)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(a + b + c)*(a + b + c)*(a + b - c)
Exercise 41, problem 27, p. 88
$x^{2} - (a + b)x + ab$ and $x^{2} - (a + c)x + ac$.
Printed answer:- $(x - a)(x - b)(x - c)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(x - a)*(x - b)*(x - c)
Exercise 41, problem 28, p. 88
$(a + b)^{2} - c^{2}$ and $a^{2} + ab + ac$.
Printed answer:- $a(a + b + c)(a + b - c)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passesa*(a + b + c)*(a + b - c)
Exercise 41, problem 3, p. 88
$4a^{3}b$ and $10ab^{3}$.
Printed answer:- $20a^{3}b^{3}$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes20*a**3*b**3
Exercise 41, problem 4, p. 88
$6a^{3}b^{3}$ and $15a^{2}b^{4}$.
Printed answer:- $30a^{3}b^{4}$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes30*a**3*b**4
Exercise 41, problem 5, p. 88
$21xy^{3}$ and $27x^{3}y^{5}$.
Printed answer:- $189x^{3}y^{5}$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes189*x**3*y**5
Exercise 41, problem 6, p. 88
$xy^{3}z^{2}$ and $x^{2}y^{2}z^{3}$.
Printed answer:- $x^{2}y^{3}z^{3}$.
verified: the printed answer passed a computed check
How it was checked
lcm: passesx**2*y**3*z**3
Exercise 41, problem 7, p. 88
$a^{2}$ and $a^{2} + a$.
Printed answer:- $a^{2}(a + 1)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passesa**2*(a + 1)
Exercise 41, problem 8, p. 88
$x^{2}$ and $x^{3} - 3x^{2}$.
Printed answer:- $x^{2}(x - 3)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passesx**2*(x - 3)
Exercise 41, problem 9, p. 88
$x^{2} - 1$ and $x^{2} + x$.
Printed answer:- $x(x + 1)(x - 1)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passesx*(x + 1)*(x - 1)