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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