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6. Answers may vary. You can use a graphing calculator to check the sketch.
The multiplicity of a zero can tell you whether the graph will cross thex-axis
at the zero or whether the zero will be a turning point at thex-axis.
7. Tonya should have factored the polynomial after finding a zero from the
graph. The function has one real root at but also two complex roots
at and.
8. Answers may vary. Sample: Graph both functions on the same grid or on a
graphing calculator to determine whether one function represents the
correct factorization of the other function. If the factorization is incorrect,
the functions will have different graphs.
9. When a graph has a multiplicity of a zero that is even, the graph only
touches the x-axis, and turns back without crossing. That never occurs in
the graph of this polynomial function.
x = − 1 x = − 1 + 2 i x = − 1 − 2 i
f (x) = x 3 + 3 x 2 + 7 x + 5
x + 1
x 2 + 2 x + 5 x 3 + 3 x 2 + 7 x + 5 (x 3 + x 2 ) ––––––––––– 2 x 2 + 7 x (2x 2 + 2 x) –––––––––––– 5 x + 5 ( 5 x + 5 ) –––––––––
x =
x = x = − 1 ± 2 i
−( 2 )±√(^2 )^2 −^4 (^1 )(^5 ) 2 ( 1 ) − 2 ±√− 16 2
10. Two real zeros at 2 and 2;
g(x) = x 4 − 16 = (x 2 − 4 )(x 2 + 4 ) = (x − 2 )(x + 2 )(x 2 + 4 ) = (x − 2 )(x + 2 )(x − 2 i)(x + 2 i) x = 2 , − 2 , 2 i, − 2 i
x 3 − 2 x 2 − 16 x + 20 = − 12 x 3 − 2 x 2 − 16 x + 32 = 0 (x 2 − 16 )(x − 2 ) = 0 (x + 4 )(x − 4 )(x − 2 ) = 0 x = − 4 , 2 , 4
f (− 4 ) = (− 4 )^3 − 2 (− 4 )^2 − 16 (− 4 ) + 32 = − 12
f ( 2 ) = ( 2 )^3 − 2 ( 2 )^2 − 16 ( 2 ) + 32 = − 12
f ( 4 ) = ( 4 )^3 − 2 ( 4 )^2 − 16 ( 4 ) + 32 = − 12
14. 0, 4; The graph crosses thex-axis at 0, and it touches thex-axis at 4.
15. −5, 1, 5; The graph crosses thex-axis at −5 , 1, and 5.
16. ; The graph crosses thex-axis at each zero.
17. The zeros of the polynomial function are: 3, −1 +i ,and − 1 − i.
f (x) = x 3 − 8 x 2 + 16 x = x(x 2 − 8 x + 16 ) = x(x − 4 )^2
g(x) = x 3 − x 2 − 25 x + 25 = (x 2 − 25 )(x − 1 ) = (x + 5 )(x − 5 )(x − 1 )
± 23 ; ± 4
f (x) = 9 x 4 − 40 x 2 + 16 0 = ( 9 x 2 − 4 )(x 2 − 4 ) = ( 3 x − 2 )( 3 x + 2 )(x 2 − 4 ) = ( 3 x − 2 )( 3 x + 2 )(x − 2 )(x + 2 ) x = ± 23 ; ± 4
f (x) = x 3 − x 2 − 4 x − 6
x − 3
x 2 + 2 x + 2 x 3 − x 2 − 4 x − 6 (x 3 − 3 x 2 ) –––––––––––– 2 x 2 − 4 x (2––––––––––––x 2 − 6 x) 2 x − 6 ( 2 x − 6 ) –––––––––
x =
x = x = − 1 ± i
−( 2 )±√(^2 )^2 −^4 (^1 )(^2 ) 2 ( 1 ) − 2 ±√− 4 2
18. Since the function is positive between the zeros and ,
Waterworks should produce between 4,000 and 12,000 paddleboards.
Zeros:
x = 4 x = 12
P(x) = − 3 x 3 + 48 x 2 − 144 x 0 = − 3 x(x 2 − 16 x + 48 ) = − 3 x(x − 4 )(x − 12 ) x = 0 , 4 , 12
− 3 x 3 − x 2 + 54 x − 40 = 2 x 2 + 6 x + 20 − 3 x 3 − 3 x 2 + 48 x − 60 = 0 − 3 (x 3 + x 2 − 16 x + 20 ) = 0 − 3 (x − 2 )^2 (x + 5 ) = 0 x = − 5 , 2
2 x 3 + 3 x 2 − 36 = x 3 − x 2 + 9 x x 3 + 4 x 2 − 9 x − 36 = 0 (x + 4 )(x + 3 )(x − 3 ) = 0 x = − 4 , − 3 , 3
− 5 x 4 + 4 x 2 − 12 x = − 6 x 4 + 3 x 3 x 4 − 3 x 3 + 4 x 2 − 12 x = 0 x(x 3 − 3 x 2 + 4 x − 12 ) = 0
x − 3
x 2 + 4 x 3 − 3 x 2 + 4 x − 12 (––––––––––––x 3 − 3 x 2 ) 4 x − 12 (4x − 12 ) ––––––––––
x(x − 3 )(x 2 + 4 ) = 0 x(x − 3 )(x + 2 i)(x − 2 i) = 0 x = − 3 , 0 , − 2 i, 2 i
Since the function is negative when , the solutions are all real
numbers such that.
25. a. A reasonable domain is , because that is where the function is
positive.
b. 0 and 10; The zeros represent the time, in seconds, when the firework is
on the ground.
c. The vertex is (5, 122.5) which represents a maximum height of 122.
meters after 5 seconds.
x > − 16
64 x 2 > − 4 x 3 − x − 16 0 = −( 4 x 3 + 64 x 2 + x + 16 )
x + 16
4 x 2 + 1 4 x 3 + 64 x 2 + x + 16 ( 4 x 3 + 64 x 2 ) ––––––––––––––– x + 16 (–––––––––x + 16 )
0 > −( 4 x 3 + 64 x 2 + x + 16 ) 0 = −(x + 16 )( 4 x 2 + 1 ) = −(x + 16 )( 2 x + i)( 2 x − i) x = − 16 , − 12 i, 12 i
x < − 16 x > − 16
0 < t ≤ 10
h = −4.9t 2 + 49 t 0 = −4.9t(t − 10 ) t = 0 , 10
26. Answers may vary. Sample: The baseball hits the ground a little past 2
seconds after being thrown. However, it is not being thrown from ground
level, but from a height of 6.5 feet. The graph only shows one zero because
the grid is only showing the parts of the graph wheret andh(t) are both
positive.
27. a.
b. , ,and
c. x − 1 represents the height of the box andx + 3 represents the length.
d. Graph the function and find the x-value when the value of
the function is 10.
, so width = 2 ft
so height = 1 ft
. length = 5 ft
28. x-axis; (x 4)
29. C
x(x + 3 )(x − 1 )
x = − 3 x = 0 x = 1
x(x + 3 )(x − 1 )
x = 2
x − 1 = 1
x + 3 = 5
