answer with complete solution

Use special products to find the product of the following: 1. (4x3 + 3) (4x3 - 3) 2. ex - y (x - 3y) 3. (xty - z)3A.) SPECIAL PRODUCTS - A shortcut in multiplying polynomials. Monomial and a Binomial - The product of a monomial and a binomial is the monomial multiplied to each term of the binomial. a(x + y) = ax + ay Illustration: a.) 4a(2x + 3y) = 4a(2x) + 4a(3y) b.) 2b(3x - 2y) = 2b(3x) + 2b(-2y) = 8ax + 12ay = 6bx - 4by 1.) Product of Sum and Difference of two terms - The product of the sum and difference of two terms is the square of the first term minus the square of the second term. (x+ y) (x- y) =x2 -y2 Illustration: a.) (3x+5y) (3x - 5y) = (3x)2 - (5y)2 b. ) (x - 2) (x+ 2) =x2-22 = 9x2 - 25y2 = x2 - 4 C.) (38) (22) = (30 + 8) (30 - 8) d.) (-2x - 5y)(2x - 5y) = -(2x + 5y) (2x - 5y) = 302 - 82 = -[(2x)2 - (5y)2] = 900 - 64 = -(4x2 - 25y2) = 836 = -4x2 + 25y2 Prepared by: RDD Department of Mathematics and Physics ENGR 1110: Mathematics of Engineers College of Sciences, CLSU Lecture Notes 2.) Square of a Binomial - The product of a square of a binomial is the square of the first term plus or minus twice the product of first and the second term plus the square of the second term. (xty)2 = x2 + 2xy+ 32 Illustration: a.) Expand (4x + 5y)2 (4x + 5y)2 = (4x)2 + 2(4x) (5y) + (5y)2 = 16x2 + 40xy + 25y2 b.) Expand (2a - 3b) 2 (2a - 3b)2 = (2a)2 - 2(2a) (3b) + (-3b)2 = 4a2 - 12ab + 962 c.) Expand (#+ 7)2 ( # + 7 ) = ( # ) + 2 ( # ) ( 7 ) + 72 -+ 7x +496.) Cube of a Trinomial - The cube of a trinomial can be treated as the cube of binomial. Illustrations: Let us consider expanding the expression (x + y + z)3. Treat (x + y) as the first term for a binomial then we proceed to obtaining the cube of a binomial. (xty+z)3 = [(x+y)+z]3 = (x+ y)3 + 3(x+ y)2(z)+ 3(x+y)(z2)+23 Prepared by: RDD CENTA Department of Mathematics and Physics ENGR 1110: Mathematics of Engineers College of Sciences, CLSU Lecture Notes = (x3 + 3x2y + 3xy2 + y3) + 3(x2 + 2xy + y2)z + (3x + 3y)(z2)+23 = x3 + 3x2y + 3xy2 + y3 + 3x2z+ 6xyz + 3y2z+ 3xz2+ 3yz2+23 =x3 + y3 + 23 + 3xly+ 3x2z+ 3xy2 + 3yz+ 3xz2+ 3yz2 + 6xyz Therefore, (xty+z)3 = x3 + y3 + 23 + 3x?y + 3x2z + 3xy2 + 3y2z+ 3xz2 + 3yz2+6xyz 7.) Binomial and Special Trinomial (x - y)(x2 +xy+ yz) = x3-y3 (x+ y)(x2 - xy+ y2) =x3+y3 What makes the trinomial special is the fact that its terms are related to the binomial in the following way: 1. two of its terms (x2 and y') are the squares of the terms in the binomial 2. the remaining term (-xy or +xy) is the negative of the product of the terms in the binomial. The product is then easily found by cubing the terms in the binomial. This product is called the Sum or Difference of Two Cubes. Illustrations: 1. (x - 3)(x2 + 3x + 9) - 33 = x3 - 27 2. (2x - 5y) (4x2 + 10xy + 25y?) = (2x)3 - (5y)3 = 8x3 - 125y3 3. (4x y + 3z?) (16x y2 - 12x2yz2 + 924) = (4x2y)3+ (3z2)3 = 64x6y3 + 27263.1) The Product of two Binomials of the form (x + a)(x + b)- The product of two binomials of the form(x + a) (x + b) is the square of x plus x times the sum of a and b plus the product of a and b. (x + a) (x+ b) = x2 + (a+ b)x+ ab Illustration Find the product of the following: a.) (x + 3) (x +8) (x + 3)(x+8) = (x)2+ (3+8)x+ (3)(8) = x2 + 11x + 24 b . ) ( x - 4) ( x + 4 ) (x - 2) (x + 4 ) =x2 + (-5+4) * +( -7)(4) =x2 + _ -1+8) x - 2 2 7 =x+=x-2 3.2) The Product of two Binomials of the form (ax + b) (cx + d)- The product of two binomials of the form (ax + b) (cx + d) is: (ax + b) (cx + d) = acx2 + (ad + bc)x + bd Illustrations: Find the product of the following: a.) (5x + 2) (4x + 3) (5x + 2) (4x + 3) = (5)(4)x2 + [5(3) + 2(4)]x + (2)(3) = 20x2 + 23x + 6 Prepared by: RDD Department of Mathematics and Physics ENGR 1110: Mathematics of Engineers College of Sciences, CLSU Lecture Notes b.) (2x - 3y) (x + 2y) (2x - 3y) (x + 2y) = 2x2 + [(2)(2y) + (-3)(1) ]x+ (-3y)(2y) = 2x2 + (4y - 3y)x - 3y2 = 2x2 + xy - 6y2 4.) Cube of a Binomial - The product of a cube of a binomial is the cube of the first term of the binomial plus or minus 3 times the square of the first term times the second term plus 3 times the first term and the square of the second term plus or minus the cube of the second term. (xty)3 = x3 +3xly+ 3xy2 + y3 Illustration: Expand the following: a.) (3x + 2y)3 (3x + 2y)3 = (3x)3 + 3(3x)2(2y) + 3(3x)(2y)2 + (2y)3 = 27x3 + 18x2y + 18xy2 + 8y3 b.) (5x - 3)3 (5x - 3)3 = (5x)3 - 3(5x)2(3) + 3(5x)(3)2 - (3)3 = 125x3 - 225x2 + 135x - 27 5.) Square of a Trinomial - To square a trinomial, get the sum of the square of each term plus twice the product of each term by another. That is, (x+ y+ z)2 = x2 + y? + z2 + 2xy+ 2xz+2yz Illustration: Expand the following: a.) (2x + 3y + 2) (2x + 3y + 2)2 = (2x)2 + (3y)2+(2)2 + 2(2x)(3y) +2(2x)(2)+2(3y)(2) = 4x2 + 9y2 + 4+ 12xy + 8x + 12y = 4x2 + 9y2 + 12xy + 8x + 12y +4 b.) (x + 2y - 4)2 (x + 2y - 4)2 = (x)2+ (2y)2+ (-4)2+2(x)(2y)+2(x)(-4)+2(2y)(-4) = x2 + 4y2 + 16 + 4xy - 8x - 16y = x2 + 4y2 +4xy - 8x - 16y + 16Monomial and a Binomial - The product of a monomial and a binomial is the monomial multiplied to each term of the binomial. a(x+ y) = ax+ ay Illustration: a.) 4a(2x + 3y) = 4a(2x) + 4a(3y) b.) 2b(3x - 2y) = 2b(3x) + 2b(-2y) = 8ax + 12ay = 6bx - 4by 1.) Product of Sum and Difference of two terms - The product of the sum and difference of two terms is the square of the first term minus the square of the second term. (x+ y) (x-y) =x2 -yz Illustration: a.) (3x+5y) (3x - 5y) = (3x)2 - (5y)2 b. ) (x - 2) (x+ 2) = x2-22 = 9x2 - 25y2 = x2 - 4 C.) (38) (22) = (30 + 8)(30 - 8) d.) (-2x - 5y) (2x - 5y) = -(2x + 5y)(2x - 5y) = 302 - 82 = -[(2x)2 - (5y)2] = 900 - 64 = -(4x2 - 25y2) = 836 = -4x2 + 25y2 Prepared by: RDD Department of Mathematics and Physics ENGR 1110: Mathematics of Engineers College of Sciences, CLSU Lecture Notes 2.) Square of a Binomial - The product of a square of a binomial is the square of the first term plus or minus twice the product of first and the second term plus the square of the second term. (xty)2 = x2 + 2xy + y2 Illustration: a.) Expand (4x + 5y)2 (4x + 5y)2 = (4x)2 + 2(4x) (5y)+ (5y)2 = 16x2 + 40xy + 25y2 b.) Expand (2a - 3b) 2 (2a - 3b)2 = (2a)2 - 2(2a) (3b) + (-3b)2 = 4a2 - 12ab + 962 c.) Expand (+ 7)2 ( 2 + 7 ) = () + 2 (# ) (7) + 72 + 7x + 49