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added algebraic quantity arithmetical progression binomial factors changing the signs clearing of fractions coefficients cube root degree denote derived polynomial dividend division dollars EXAMPLES FOE EXAMPLES FOR PRACTICE exponent expression figure Find the cube Find the sum find the values FOE PRACTICE following Rule formula fourth geometrical progression geometrical series given equation given number given quantities greater greatest common divisor identical equation imaginary indicated inequality irreducible fraction last term least common multiple less letters minus sign monomial Multiply negative quantity nth root number of terms obtain OPERATION partial fractions permutations positive roots problem proportion quadratic quadratic equation quotient radical sign rational Reduce remainder represent required root result scries second member second term square root Sturm's Theorem subtracted suppose surd taken third three numbers tion tlie transformed equation transposing trial divisor unity unknown quantity whence whole number zero
Page 216 - ... the product of the two, plus the square of the second. In the third case, we have (a + b) (a — 6) = a2 — b2. (3) That is, the product of the sum and difference of two quantities is equal to the difference of their squares.
Page 73 - To reduce a fraction to its lowest terms. A fraction is in its lowest terms, when the numerator and denominator are prime to each other.
Page 408 - VARIATIONS of sign, nor the number of negative roots greater than the number of PERMANENCES. 325. Consequence. When the roots of an equation are all real, the number of positive roots is equal to the number of variations, and the number of negative roots is equal to the number of permanences. For, let m denote the degree of the equation, n the number of variations of the signs, p the number of permanences ; we shall have m=n+p. Moreover, let n' denote the number of positive roots, and p' the number...
Page 185 - Multiply the divisor, thus increased, by the last figure of the root; subtract the product from the dividend, and to the remainder bring down the next period for a new dividend.
Page 371 - From this we might conclude that every equation involving but one unknown quantity, has as many roots as there are units in the exponent of its degree, and can have no more.
Page 84 - Reduce compound fractions to simple ones, and mixt numbers to improper fractions ; then multiply the numerators together for a new numerator, and the denominators for. a new denominator.
Page 19 - Fractional exponents are used to denote both involution and evolution in the same expression, the numerator indicating the power to which the quantity is to be raised, and the denominator the required root of this power. Thus, the expression a* signifies the 4th root of the 3d power of a, and is equivalent to Va'.
Page 43 - The square of the sum of two quantities is equal to the square of the first, plus twice the product of the first and second, plus the square of the second.
Page 105 - Divide 48 into two such parts, that if the less be divided by 4, and the greater by 6, the sum of the quotients will be 9. Ans. 12 and 36. 11. An estate is to be divided among 4 children, in the following manner : The first is to have $200 more than 1 of the whole.