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Simplify, (a – x)(b – x)(c – x) . . . (z –x) Simplify, (a – x) . . . (x – x)(y – x)(z –x) Simplify, (a – x) . . . (x – x)(y – x)(z –x) Simplify, (a – x) . . . (x – x)(y – x)(z –x) (a – x) . . . (0)(y – x)(z –x) Simplify, (a – x) . . . (x – x)(y – x)(z –x) (a – x) . . . (0)(y – x)(z –x) Any number times zero is zero. Simplify, (a – x) . . . (x – x)(y – x)(z –x) (a – x) . . . (0)(y – x)(z –x) Any number times zero is zero. 0 x2 + 9x + 20 = 0 x2 + 9x + 20 = 0 (x + 4)(x + 5) = 0 x2 + 9x + 20 = 0 (x + 4)(x + 5) = 0 (x + 4) = 0 (x + 5) = 0 x2 + 9x + 20 = 0 (x + 4)(x + 5) = 0 (x + 4) = 0 (x + 5) = 0 x = -4 x2 + 9x + 20 = 0 (x + 4)(x + 5) = 0 (x + 4) = 0 (x + 5) = 0 x = -4 or x = -5 The Zero Product Property: If two or more numbers have a product of zero, then at least one of the numbers must be zero, or, for any numbers a and b, if ab = 0, then either a = 0 or b = 0. 3x2 + 15x = 0 3x2 + 15x = 0 3x(x + 5) = 0 3x2 + 15x = 0 3x(x + 5) = 0 3x = 0 3x2 + 15x = 0 3x(x + 5) = 0 3x = 0 (x + 5) = 0 3x2 + 15x = 0 3x(x + 5) = 0 3x = 0 x=0 (x + 5) = 0 3x2 + 15x = 0 3x(x + 5) = 0 3x = 0 x=0 (x + 5) = 0 x = -5 x – 16 = 0 2 x – 16 = 0 2 (x – 4)(x + 4) = 0 x – 16 = 0 2 (x – 4)(x + 4) = 0 (x – 4) = 0 x – 16 = 0 2 (x – 4)(x + 4) = 0 (x – 4) = 0 (x + 4) = 0 x – 16 = 0 2 (x – 4)(x + 4) = 0 (x – 4) = 0 x=4 (x + 4) = 0 x – 16 = 0 2 (x – 4)(x + 4) = 0 (x – 4) = 0 (x + 4) = 0 x = 4 or x = -4 x3 – 6x2 – 16x = 0 x3 – 6x2 – 16x = 0 x(x2 – 6x – 16) = 0 x3 – 6x2 – 16x = 0 x(x2 – 6x – 16) = 0 x(x – 8)(x + 2) = 0 x3 – 6x2 – 16x = 0 x(x2 – 6x – 16) = 0 x(x – 8)(x + 2) = 0 x=0 x3 – 6x2 – 16x = 0 x(x2 – 6x – 16) = 0 x(x – 8)(x + 2) = 0 x = 0 (x – 8) = 0 x3 – 6x2 – 16x = 0 x(x2 – 6x – 16) = 0 x(x – 8)(x + 2) = 0 x = 0 (x – 8) = 0 (x + 2) = 0 x3 – 6x2 – 16x = 0 x(x2 – 6x – 16) = 0 x(x – 8)(x + 2) = 0 x = 0 (x – 8) = 0 (x + 2) = 0 x=0 x3 – 6x2 – 16x = 0 x(x2 – 6x – 16) = 0 x(x – 8)(x + 2) = 0 x = 0 (x – 8) = 0 (x + 2) = 0 x = 0 or x = 8 x3 – 6x2 – 16x = 0 x(x2 – 6x – 16) = 0 x(x – 8)(x + 2) = 0 x = 0 (x – 8) = 0 (x + 2) = 0 x = 0 or x = 8 or x = -2 7x2 – 21x – 28 = 0 7x2 – 21x – 28 = 0 7(x2 – 3x – 4) = 0 7x2 – 21x – 28 = 0 7(x2 – 3x – 4) = 0 7(x + 1)(x – 4) = 0 7x2 – 21x – 28 = 0 7(x2 – 3x – 4) = 0 7(x + 1)(x – 4) = 0 (x + 1) = 0 7x2 – 21x – 28 = 0 7(x2 – 3x – 4) = 0 7(x + 1)(x – 4) = 0 (x + 1) = 0 (x – 4) = 0 7x2 – 21x – 28 = 0 7(x2 – 3x – 4) = 0 7(x + 1)(x – 4) = 0 (x + 1) = 0 (x – 4) = 0 x = -1 7x2 – 21x – 28 = 0 7(x2 – 3x – 4) = 0 7(x + 1)(x – 4) = 0 (x + 1) = 0 (x – 4) = 0 x = -1 or x = 4 x2 – 14x + 49 = 0 x2 – 14x + 49 = 0 (x – 7)(x – 7) = 0 x2 – 14x + 49 = 0 (x – 7)(x – 7) = 0 (x – 7) = 0 x2 – 14x + 49 = 0 (x – 7)(x – 7) = 0 (x – 7) = 0 (x – 7) = 0 x2 – 14x + 49 = 0 (x – 7)(x – 7) = 0 (x – 7) = 0 (x – 7) = 0 x= 7 16x2 – 36 = 0 16x2 – 36 = 0 4(4x2 – 9) = 0 16x2 – 36 = 0 4(4x2 – 9) = 0 4(2x – 3)(2x + 3) = 0 16x2 – 36 = 0 4(4x2 – 9) = 0 4(2x – 3)(2x + 3) = 0 2x – 3 = 0 16x2 – 36 = 0 4(4x2 – 9) = 0 4(2x – 3)(2x + 3) = 0 2x – 3 = 0 2x + 3 = 0 16x2 – 36 = 0 4(4x2 – 9) = 0 4(2x – 3)(2x + 3) = 0 2x – 3 = 0 2x = 3 2x + 3 = 0 16x2 – 36 = 0 4(4x2 – 9) = 0 4(2x – 3)(2x + 3) = 0 2x – 3 = 0 2x = 3 x = 1.5 2x + 3 = 0 16x2 – 36 = 0 4(4x2 – 9) = 0 4(2x – 3)(2x + 3) = 0 2x – 3 = 0 2x = 3 x = 1.5 2x + 3 = 0 2x = -3 16x2 – 36 = 0 4(4x2 – 9) = 0 4(2x – 3)(2x + 3) = 0 2x – 3 = 0 2x = 3 x = 1.5 or 2x + 3 = 0 2x = -3 x = -1.5 (18x3 – 50x)(x2 + 15x + 54) = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 3x – 5 = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 3x + 5 = 0 3x – 5 = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 3x + 5 = 0 x+9=0 3x – 5 = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 3x + 5 = 0 x+6=0 x+9=0 3x – 5 = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 x=0 3x + 5 = 0 x+6=0 x+9=0 3x – 5 = 0 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 x=0 3x + 5 = 0 x+6=0 x+9=0 3x – 5 = 0 3x = 5 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 3x + 5 = 0 x+6=0 x+9=0 3x – 5 = 0 x=0 3x = 5 3x = 5/3 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 x=0 3x + 5 = 0 x+6=0 x+9=0 3x – 5 = 0 3x = -5 3x = 5 3x = 5/3 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 x=0 3x + 5 = 0 x+6=0 x+9=0 3x – 5 = 0 3x = -5 3x = 5 3x = 5/3 3x = -5/3 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 x=0 3x + 5 = 0 x+6=0 x+9=0 3x – 5 = 0 3x = -5 3x = 5 3x = 5/3 3x = -5/3 x = -9 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 x=0 3x + 5 = 0 x+6=0 x+9=0 3x – 5 = 0 3x = -5 3x = 5 3x = 5/3 3x = -5/3 x = -9 x = -6 (18x3 – 50x)(x2 + 15x + 54) = 0 2x(9x2 – 25)(x + 9)(x + 6) = 0 2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0 2x = 0 x=0 3x + 5 = 0 x+6=0 x+9=0 3x – 5 = 0 3x = -5 3x = 5 3x = 5/3 x = -9 x = -6 3x = -5/3 x = -9, -6, -5/3, 0, 5/3 x2 – 15x = 50 – 10x x2 – 15x = 50 – 10x x2 – 5x – 50 = 0 x2 – 15x = 50 – 10x x2 – 5x – 50 = 0 (x – 10)(x + 5) = 0 x2 – 15x = 50 – 10x x2 – 5x – 50 = 0 (x – 10)(x + 5) = 0 (x – 10) = 0 x2 – 15x = 50 – 10x x2 – 5x – 50 = 0 (x – 10)(x + 5) = 0 (x – 10) = 0 (x + 5) = 0 x2 – 15x = 50 – 10x x2 – 5x – 50 = 0 (x – 10)(x + 5) = 0 (x – 10) = 0 (x + 5) = 0 x = 10 x2 – 15x = 50 – 10x x2 – 5x – 50 = 0 (x – 10)(x + 5) = 0 (x – 10) = 0 (x + 5) = 0 x = 10 or x = -5 4x3 + x2 + 20 = 4x3 – 12x 4x3 + x2 + 20 = 4x3 – 12x 4x3 + x2 + 20 = 4x3 – 12x x2 + 12x + 20 = 0 4x3 + x2 + 20 = 4x3 – 12x x2 + 12x + 20 = 0 (x + 10)(x + 2) = 0 4x3 + x2 + 20 = 4x3 – 12x x2 + 12x + 20 = 0 (x + 10)(x + 2) = 0 (x + 10) = 0 4x3 + x2 + 20 = 4x3 – 12x x2 + 12x + 20 = 0 (x + 10)(x + 2) = 0 (x + 10) = 0 (x + 2) = 0 4x3 + x2 + 20 = 4x3 – 12x x2 + 12x + 20 = 0 (x + 10)(x + 2) = 0 (x + 10) = 0 (x + 2) = 0 x = -10 4x3 + x2 + 20 = 4x3 – 12x x2 + 12x + 20 = 0 (x + 10)(x + 2) = 0 (x + 10) = 0 (x + 2) = 0 x = -10 or x = -2 Some equations can be solved using factoring. First perform some transformations so that one side of the equation is equal to zero. Then factor the other side of the equation. Use the Zero Product Property to find all of the solutions to the equation.

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