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Zeros of a Function The zeros of a function are the x-intercepts of the graph. Zeros are also called roots or solutions. In general, a function can have as many zeros as its highest exponent. Quadratic functions can have 0, 1, or 2 zeros. Quadratic zeros are symmetric about the axis of symmetry. No zeros: 1 zero: 2 zeros: Zeros can be found by factoring and setting the equation equal to zero. Ex) f(x) = 5x2 + 10x 5x(x – 2) = 0 Factor by GCF 5x = 0 and x – 2 = 0 Set each term = 0 x = 0 and x = 2 Solve for x 5x and (x – 2) are the factors. x = 0 and x = 2 are the zeros. Ex) g(x) = x2 – 10x + 9 (x – 9)(x – 1) Factor x – 9 = 0 and x – 1 = 0 Set each term = 0 x = 9 and x = 1 Solve for x Ex) x2 + 5x = -6 x2 + 5x + 6 = 0 (x + 2)(x + 3) = 0 x + 2 = 0 and x + 3 = 0 x = -2 and x = -3 Get equation equal to 0 Factor Set each term = 0 Solve for x Ex) 3x2 – 8x = x – 6 3x2 – 9x + 6 = 0 3(x2 – 3x + 2) = 0 3(x – 2)(x – 1) = 0 x – 2 = 0 and x – 1 = 0 x=2 and x = 1 Get equation equal to 0 Factor the GCF Factor the parenthesis Set each term = 0 Solve for x Ex) 5x2 = 125 5x2 – 125 = 0 5(x2 – 25) = 0 5(x + 5)(x – 5) = 0 x + 5 = 0 and x – 5 = 0 x = -5 and x = 5 Get equation equal to 0 Factor the GCF Factor the parenthesis Set each term = 0 Solve for x Ex) g(x) = x2 + 6x + 9 (x + 3)(x + 3) Factor x + 3 = 0 and x + 3 = 0 Set each term = 0 x = -3 and x = -3 Solve for x x = -3 is a double root Ex) 3x2 + 8x = -4 3x2 + 8x + 4 = 0 (x + 2)(x + 6) 2 6 (x + )(x + ) Get equation equal to 0 Bottom’s Up: ac = 12 Divide by leading coeff. 3 3 2 Reduce each fraction (x + )(x + 2) 3 Bring the bottom up (3x + 2)(x + 2) 3x + 2 = 0 and x + 2 = 0 Set each factor = 0 Solve for x x = -3/2 and x = -2 Ex) 2x2 + 12 = 11x 2x2 + 11x + 12 = 0 (x + 3)(x + 8) 3 8 (x + )(x + ) Get equation equal to 0 Bottom’s Up: ac = 24 Divide by leading coeff. 2 2 3 Reduce each fraction (x + )(x + 4) 2 Bring the bottom up (2x + 3)(x + 4) 2x + 3 = 0 and x + 4 = 0 Set each factor = 0 Solve for x x = -3/2 and x = -4 Ex) 2x3 + 8x2 = -6x 2x3 + 8x2 + 6x = 0 2x(x2 + 4x + 3) 2x(x + 3)(x + 1) 2x = 0, x + 3 = 0, x + 1 = 0 x = 0, x = -3, and x = -1 Get equation equal to 0 Factor the GCF Factor the parenthesis Set each factor = 0 Solve for x