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ence of two fourth powers, (x3) 4 — (y3)4, the difference of two cubes, (x4) 3 (y1) or the difference of two squares, (x6)2 — (y6) 2.

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All possible factors of such a binomial are most easily obtained by considering it the difference of the lowest powers.

Factoring as the difference of two squares,

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The first factor, the sum of two cubes, the second factor the difference of two squares,

=

(x + y) (x — yε) (x2 + y2)(x1 — x2y2 + y1) (x3 + y3)(x3 — y3) The last two factors, being the sum and difference of two cubes, may be factored farther.

(x2 + y2)(x4 x2y2 + y1) (x3 + y3)(x3 — y3)

=

(x2 + y2)(x − x2y2 + y1) (x + y) (x2 — xy + y2) (x − y) (x2 + xy + y2) which gives us all the prime factors of x12 — y12.

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69. Case 11. Theory of Divisors or Factor Theorem.

If a polynominal in x, as x3- 2x2 + 4x8 is divided by 2, the quotient will be x2 + 4.

x

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to zero for x = 2, then x 2 is a factor of the expression.

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Hence the Factor Theorem:

n is a factor of the

If a polynomial in x is reduced to zero when a particular number n is substituted for x, then x polynomial.

1. Factor x5 + 4x2 + 16.

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n, it is evident

If this expression contains a factor of the form x that n must be an exact divisor of the last term, 16. The exact divisors of 16 are, 1, 1, 2, 2, 4, 4, 8,

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- 16. Since our expression contains only positive terms, no plus value could reduce it to zero. Hence we only need to try the negative divisors of 16.

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1 does not reduce, the expression to zero. Sub

(− 2)5 + 4(− 2)2 + 16
(-2) = x + 2 must be a factor.

=

32 +16 +16 = 0

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Hence x
Dividing by x + 2, the quotient is x4
Therefore, x5 + 4x2 + 16

=

As no exact divisor of 8 will reduce the second factor to zero, both factors are prime.

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CHAPTER VIII

HIGHEST COMMON FACTOR AND LEAST COMMON MULTIPLE

COMMON FACTORS

70. A common factor of two or more numbers is a factor contained in each of the numbers and therefore, an exact divisor of each number. This definition holds, whether the numbers are expressed in figures, as in Arithmetic, or expressed in figures and letters, as in Algebra. Integral and rational expressions only, that is, expressions that do not contain fractions or indicated roots, are considered in this chapter.

Thus, 3x is a common factor of 12x and 15x2, and a + b is a common factor of (a + b)2 and (a + b)(a — b).

As in Arithmetic, two or more algebraic numbers are said to be prime to each other when they have no common factor except 1.

The highest common factor of two or more algebraic numbers is the highest or largest factor common to all the numbers.

Thus, 2 is a common factor of 12 and 16, but 4 is the largest common factor.

Similarly, 3x is a common factor of 12x2 and 15x3, but 3x2 is the highest common factor, that is, the factor of highest degree contained in each number.

It is convenient to use the abbreviation H. C. F. for highest common factor.

1. Find the H. C. F. of 28x2y3 and 35x3y5.

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2. Find the H. C. F. of x3 + y3 and x2 + 2xy + y2.

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3. Find the H. C. F. of 6a2 - 12ab+ 6b2; 9a2 - 9b2; 12a2 + 24ab

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36b2.

12ab+6b2 = 6(a2
· 9(a2 — b2)

-962 =

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=

Hence, to find the H. C. F. of two or more algebraic expressions, use the following

Rule.-Factor each expression.

The product of all the common factors is the highest common factor.

Note, that each common factor is used the least number of times it occurs in any expression.

EXERCISE 44

Find the H. C. F. of:

1. 12x3 and 30x2.

2. 2a3bc and a2b2c.

3. 50cx3y2, 25cx1y, and 50cx3. 4. 42ak5, 21a2k6 and 28a3k8.

5. 24m3n3, 42a2m2n2y, and 36m2n4.

6. 2(xy)2 and 4(x − y)3.

3

7. 6c(a + b)3 and 12b(a + b)5.

8. 10(x + y)(x − y)2 and 15a(x + y)2(x − y).

9. 7ab(m + n)2(m — n), 21a2b2(m + n) (m − n)2 and 14a3b(m + n)3(m — n)3.

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