sion is not in the required form, see if it can be brought under any of the given types, especially in cases of binomials and trinomials of the second degree.

3. If the expression cannot be brought under any of the given types, the method of the Remainder Theorem should be tried to test for binomial factors.

SOLUTION OF EQUATIONS BY FACTORING

160. Factoring finds a simple and valuable application in the practical solution of many equations of higher degree than the first. The use of the factoring method in solving equations is shown in the examples considered below.

161. As we have seen (Art. 30), to solve an equation is to find the value of the unknown number - the root of the equation-which when substituted in the original equation converts it into an identity.

Thus, 4 is such a value of x in the equation 2x+3= 11. Substituting 4 for x, 2 x 4 + 3 = 11.

1. Find the value of x in the equation a2+3 x = 10.

The first member evidently reduces to 0, if either factor is 0, because the product will then be 0 (Art. 93).

Putting x 20, then x = 2, by adding 2.

Putting x + 5 = 0, then x = - -5, by subtracting 5.

.. x=2 or -5; for if x = 2, then 2 x 2 + 3 × 2 − 10 = 0; and if x=-5, then 5 x - 5 + 3 × 5 - 10

=

0.

2. Find the value of a in the equation a2 - 9 = 0.

x2 - 9 = 0.

(x+3)(x − 3) = 0, by factoring.

The first member evidently reduces to 0, if either factor is 0. Why?

3. Solve the equation a3 - 2x2 − 5 x+6= 0.

x3- 2x2 - 5 x + 6 = 0.

(Why?) (Why ?)

(x − 1)(x2 x 6)= 0, by method of Rem. Theorem.

The first member of this equation will evidently become equal to the second (i.e., the equation will reduce to an identity), if any factor of the first member equals 0. Why?

16. x2-7x=-12.

17. 2y-y-3=0.

18. 65 - x2 = 16.

19. 6x+x-15=0.
20. y2+4y=32.

21. 3-9x+27 x=27.
22. xx2-x+1=0.
23. 1-3x+3x2x2=0.
24. x2-3x-40 ao - 1.
25. (2x+3)(3x+4)=0.
26. 2-6 +11 x2-6x=0.
27. x3-4x2+10x=3x2-2 x.

=

28. If 6 is added to the square of a number, the sum will be equal to 5 times the number. Find the number.

29. If 27 is subtracted from the square of a number, the remainder will be 6 times the number. Find the number.

30. If 20 is added to a certain number, and 5 is subtracted from the same number, the product of the sum and difference thus obtained will be equal to -100. Find the number.

31. The number of square rods in the area of a square field, diminished by 40, is equal to 3 times the number of rods in a side. How many rods in a side of the field?

32. The square of the number expressing the number of dollars a man is worth is greater by 165 than 4 times that number. How much is he worth? What does the negative result mean here?

33. If the square of a number and the number itself be subtracted from the cube of the number, the remainder will be 5 times the number. Find the number (or numbers).

HIGHEST COMMON FACTOR

162. The Highest Common Factor of two or more algebraic expressions is the algebraic factor of highest degree which is common to them. It will of course exactly divide each of them.

Thus, a is a common factor of abs and ab2c; but the highest common factor is ab2. The highest common factor of x2 - 1 and x2 + 3x + 2 is x+1, which is the only common factor; the factors of x2 1 being x + 1 and x − 1, and those of x2 + 3x + 2 being x + 1 and x + 2.

163. The highest common factor has reference only to the degree of the expression and should not be confused with the greatest common divisor in arithmetic. They are not necessarily the same when numerical values are assigned to the letters contained in the expressions. For example, the highest common factor of a2 and ab2 is a; but if a = 2 and b = 4, the greatest common divisor of 22 and 2 × 42 is 4, and not a, or 2.

NOTE. Since common numerical factors have nothing to do with the algebraic divisibility of the expressions and do not affect the degree of the highest common factor, it is not necessary to consider them so far as the algebraic highest common factor is concerned. It is customary, however, in many text-books to write the greatest common divisor of the numerical factors as a coefficient of the algebraic highest common factor.

164. By Factoring Method.

When the given expressions can be easily factored, their highest common factor is usually found by inspection. It is necessary merely to factor the expressions and then take the product of all the common factors, giving to each the lowest exponent with which it occurs in any of the expressions.