(29.) The numerical value of an algebraic expression is the result obtained when we attribute particular values to the letters.

Suppose the expression is 2a2b.

If we make a=2 and 6-3, the value of this expression will

be 2×2×2×3=24.

If we make a=4 and b=3, the value of the same expression will be 2X4X4X3=96.

The numerical value of a polynomial is not affected by changing the order of the terms, provided we preserve their respective signs.

The expressions a'+2ab+b', a2+b2+2ab, b'+2ab+a', have all the same numerical value.

Thus, if a 5 and b=2, the value of a' will be 25, that of 2ab will be 20, and b2 will be 4; and if these numbers are added together, their sum will be the same in whatever order they are placed. Thus,

(30.) Each of the literal factors which compose a term is called a dimension of this term; and the degree of a term is the number of these factors or dimensions. A numerical coefficient is not counted as a dimension.

Thus, 3a is a term of one dimension, or of the first degree. 5ab is a term of two dimensions, or of the second degree. 6a2bc is a term of six dimensions, or of the sixth degree. In general, the degree, or the number of dimensions of a term, is equal to the sum of the exponents of the letters contained in the term.

Thus, the degree of the term 5ab'cd is 1+2+1+3 or 7 that is, this term is of the seventh degree.

(31.) A polynomial is said to be homogeneous when all its terms are of the same degree.

Thus, 2a-3b+c, is of the first degree and homogeneous.

3a-4ab+b', is of the second degree and homogeneous. 2a'+3ac-4c'd, is of the third degree and homogene

ous.

5a-2ab+c, is not homogeneous

(32.) Like or similar terms are terms composed of the same letters affected with the same exponents.

Thus, 3ab and Tab are similar terms.

5a'c and 3a'c are also similar terms.

But 3ab2 and 4a'b are not similar; for, although they contain the same letters, the same letters are not affected with the same exponents.

(33.) The reciprocal of a quantity is the quotient arising from dividing a unit by that quantity.

Thus, the reciprocal of 2 is; the reciprocal of a is

(34.) A few examples are here subjoined, to exercise the learner on the preceding definitions and remarks.

Examples in which words are to be converted into algebraic symbols.

Ex. 1. What is the algebraic expression for the following statement? The second power of a, increased by twice the product of a and b, diminished by c, and increased by d, is equal to seventeen times f

Ans. a'+2ab-c+d=17f.

Ex. 2. The quotient of three divided by the sum of x and four, is equal to twice b diminished by eight.

Ex. 3. One third of the difference between six times x and four, is equal to the quotient of five divided by the sum of a and b.

Ex. 4. Three quarters of x increased by five, is equal to three sevenths of b diminished by seventeen.

Ex. 5. One ninth of the sum of six times x and five, added to one third of the sum of twice x and four, is equal to the product of a, b, and c.

Ex. 6. The quotient arising from dividing the sum of a and b by the product of c and d, is equal to four times the sum of e, f, g, and h.

(35.) Examples in which the algebraic signs are to be trans lated into common language.

Ans. The quotient arising from dividing the sum of x and a by b, increased by the quotient of x divided by c, is equal to the quotient of d divided by the sum of a and b.

Ex. 2. 7a2+(b-c)× (d+e)=g+h.

How should the preceding example be read, when the first parenthesis is omitted?

(36.) Find the value of the following expressions, when a=

Ex. 9. What is the value of √5x+5√y+√x+vy, when x=9 and y=4?

Ex. 10. What is the value of x-4x2+7x2-6x, when x=3? Ex. 11. What is the value of 5(x+y)+4xy, when x=4 and y=6?

Ex. 12. What is the value of √10+x−√10+x, when x=6? Ex. 13. What is the value of 2x2+ √2x2+1, when x=2? Ex. 14. What is the value of 2x-7√x, when x=81?

SECTION II.

ADDITION.

(37) Addition is the connecting of quantities together by means of their proper signs, and incorporating such as can be united into one sum.

It is convenient to distinguish three Cases.

CASE I.

When the quantities are similar and have the same signs.

RULE.

Add the coefficients of the several quantities together, and to their sum annex the common letter or letters, prefixing the common sign.

Thus, the sum of 3a and 5a is obviously 8a. So, also, -3a and -5a make -8a; for the minus sign before each of the terms shows that they are to be subtracted, not from each other, but from some quantity which is not here expressed; and if 3a and 5a are to be successively subtracted from the same quantity, it is the same as subtracting at once 8a.

The learner must continually bear in mind the remark of

Art. 13, that when no sign is prefixed to a quantity, plus is always to be understood.

CASE II.

(38.) When the quantities are similar, but have different signs.

RULE.

Add all the positive coefficients together, and also all those that are negative; subtract the least of these results from the greater; to the difference annex the common letter or letters, and prefix the sign of the greater sum.

Thus, instead of 7a-4a, we may write 3a, since these two expressions obviously have the same value.

Also, if we have 5a-2a+3a-a, this signifies that from 5a we are to subtract 2a, add 3a to the remainder, and then subtract a from this last sum, the result of which operation is 5a But it is generally most convenient to take the sum of the positive quantities, which in this case is 8a; then take the sum of the negative quantities, which in this case is 3a; and we have 8a-3a or 5a, the same result as before.

(39.) When some of the quantities are dissimilar.

RULE.

Collect all the like quantities together, by taking their sums or differences as in the two former cases, and set down those that are unlike, one after the other, with their proper signs.

Unlike quantities can not be united in one term. Thus, 2a and 36 neither make 5a nor 5b. Their sum can only be written 2a+3b.