§ 6. THE SQUARE OF ANY POLYNOMIAL 95. Law. Perform the operation indicated by (2n+3b+4x3-6y2)2. In the result do you have the sum of the squares of each of the terms? Do you also have plus twice the product of each term. into each which successively follow it? Square any other polynomial and see if the result corresponds. Write the law. 96. Examples for Sight Work. By application of the law expand* the following: 11. (y+3yn-2+9yn-by-n-1) 2. 12. (7x1-6x3-2x2-5x-9)2. 13. (√d−√Ã−3x3+7x-8√x)2. 14. (ax+bx+cx+ay+dy)2. 15. (-3y-6y-7y2-2y3-3)2. 17. (√3x-√2y+3a−√4b)2 16. (ax2+br3 - 2cx — d1 —9)2. 18. (xn−x3n -2x1n — xn+1)2. * Expand means to raise to the indicated power. § 7. THE EXACT DIVISOR OF A POLYNOMIAL 97. The Factor Law. What value of x will make 3(x-7) equal to zero? If any factor of a product is zero what does the product equal? What value of x will make x-4 equal to zero? If x-4 is a factor of any expression will the expression equal zero when x-4 equals zero? In (x-3)(x-6) name two values of x which will make the expression equal to zero. What value substituted for x will reduce to zero, every expression in which x-2 is a factor? An expression having x-9 as a factor will reduce to zero when what is substituted for x? An expression having x-12 as a factor will reduce to zero when what is substituted for x? Therefore an expression having x-a as a factor will reduce to zero when what is substituted for x? Therefore if any rational integral expression containing x, does not reduce to zero when a is substituted for x, is it divisible by x-a? Law. Any rational, integral expression in x, which reduces to zero when x equals a, is exactly divisible by x-a. 98. Illustration. (1) To determine an exact divisor or factor of 5x5-6x3+1 we substitute 1 for x in the expression. Since it reduces to zero we know that it is exactly divisible by x-1. (2) To determine an exact divisor of x3+6x2+11x+6 we substitute for x, some factor of 6. But since the signs are all positive it is evident that no positive factor of 6 will reduce the expression to zero. We therefore try -3 instead of +3 and obtain the following: −27+54−33+6=0. The expression is therefore exactly divisible by x-(-3), which by the law of subtrahends becomes x+3. (3) To determine the exact divisor of x3-7x2+16x-12 we substitute for x some factor of 12, say 3. When this is substituted for x we have 27-63+48-12=0. Therefore x-3 is an exact divisor of the expression. When the known term has several factors several trials must sometimes be made before the correct factor is determined. 99. Examples. Determine an exact divisor or factor of each of the following expressions: CHAPTER VI FACTORING SECTION 1, A COMMON FACTOR. SECTION 2, GROUPING. SECTION 3, THE DIFFERENCE OF Two SQUARES. SECTION 4, THE DIFFERENCE OF TWO CUBES. SECTION 5, THE SUM OF TWo CUBES. SECTION 6, THE TRINOMIAL. SECTION 7, THE POLYNOMIAL. SECTION 8, SPECIAL EXPRESSIONS. 100. Definitions. The factors of a number or expression are the quantities whose product equals the number or expression. Factor means maker and factors are numbers which make other numbers when multiplied together. Factoring is the process of resolving an expression into its factors. To resolve an expression into its factors is to determine its factors. 101. Cases of Factoring. This subject will be presented under the following cases: § 1. A COMMON FACTOR 102. Illustration. In the expression 25x+15x3-30x2-40x+100 what is the largest factor common to all the terms? Write this factor before a parenthesis within which is the result obtained by dividing the expression by this common factor. What is the largest factor which is evenly divisible in 8a2x2-24a4x-32a3x1+16ax3. Express the factors in the same form as in the first illustration. 103. Examples. Factor the following expressions: 1. 3a2c-27a4c3+9a3c2-15a1c1. [ 2. 5ax2-3x2+cx2. In the result what is the coefficient of x2? Why? 3. ƒahy3 —ƒh2y2+f3h3y. 4. 3x-bx+cx-tx. In the result what is the coefficient of x? |