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Zeros of Polynomials
25. Start with p(x) = −6x^{3} + 4x^{2} + 16x. Factor out the
gcf (−2x in this case), then
use the ac method to complete the factorization.
p(x) = −2x[3x^{2} − 2x − 8]
p(x) = −2x[3x^{2} − 6x + 4x − 8]
p(x) = −2x[3x(x − 2) + 4(x − 2)]
p(x) = −2x(3x + 4)(x − 2)
Set
0 = −2x(3x + 4)(x − 2)
and use the zero product property to write
−2x = 0 or 3x + 4 = 0 or x − 2 = 0.
Solving , the zeros are x = 0, −4/3, and 2.
27. Start with p(x) = −2x^{7} − 10x^{6} + 8x^{5} + 40x^{4}. Factor
out the gcf (−2x^{4} in this
case), then use grouping and difference of squares to complete the
factorization.
p(x) = −2x^{4}[x^{3} + 5x^{2} − 4x − 20]
p(x) = −2x^{4}[x^{2}(x + 5) − 4(x + 5)]
p(x) = −2x^{4}(x^{2} − 4)(x + 5)
p(x) = −2x^{4}(x + 2)(x − 2)(x + 5)
Set
0 = −2x^{4}(x + 2)(x − 2)(x + 5)
and use the zero product property to write
−2x^{4} = 0 or x + 2 = 0 or x − 2 = 0 or x + 5 = 0.
Solving, the zeros are x = 0, −2, 2, and −5.
29. The graph of the polynomial
intercepts the x axis at (−4, 0), (1, 0), and (2, 0). Hence, the zeros of the
polynomial
are −4, 1, and 2.
31. The graph of the polynomial
intercepts the xaxis at (−4, 0), (0, 0), and (5, 0). Hence, the zeros of the
polynomial
are −4, 0, and 5.
33. The graph of the polynomial
intercepts the xaxis at (−3, 0), (0, 0), (2, 0), and (6, 0). Hence, the zeros
of the polynomial
are −3, 0, 2, and 6.
35. Factor p (x) = 5x^{3} + x^{2} − 45x − 9 by grouping, then
complete the factorization
with the difference of squares pattern.
p(x) = x^{2}(5x + 1) − 9(5x + 1)
p(x) = (x^{2} − 9)(5x + 1)
p(x) = (x + 3)(x − 3)(5x + 1)
Using the zero product property , the zeros are −3, 3, and
−1/5. Hence, the graph
of the polynomial must intercept the xaxis at (−3, 0), (3, 0), and (−1/5, 0).
Further,
the leading term of the polynomial is 5x^{3}, so the polynomial must have the same endbehavior
as y = 5x^{3}, namely, it must rise from negative infinity, wiggle through its
xintercepts, then rise to positive infinity . The sketch with the appropriate
zeros and
end behavior follows.
Checking on the calculator .
37. Factor p(x) = 4x^{3} − 12x^{2} − 9x + 27 by grouping, then complete the
factorization
with the difference of squares pattern .
p(x) = 4x^{2}(x − 3) − 9(x − 3)
p(x) = (4x^{2} − 9)(x − 3)
p(x) = (2x + 3)(2x − 3)(x − 3)
Using the zero product property , the zeros are −3/2, 3/2,
and 3. Hence, the graph
of the polynomial must intercept the xaxis at (−3/2, 0), (3/2, 0), and (3, 0).
Further,
the leading term of the polynomial is 4x^{3}, so the polynomial must have the same
endbehavior
as y = 4x^{3}, namely, it must rise from negative infinity, wiggle through its
xintercepts, then rise to positive infinity. The sketch with the appropriate
zeros and
end behavior follows.
Checking on the calculator.
39. Start with p(x) = x^{4} + 2x^{3} − 25x^{2} − 50x, then factor out the gcf (x in
this case).
Then, factor by grouping and complete the factorization with the difference of
squares
pattern.
p(x) = x[x^{3} + 2x^{2} − 25x − 50]
p(x) = x[x^{2}(x + 2) − 25(x + 2)]
p(x) = x(x^{2} − 25)(x + 2)
p(x) = x(x + 5)(x − 5)(x + 2)
Using the zero product property, the zeros are 0, −5, 5,
and −2. Hence, the graph of
the polynomial must intercept the xaxis at (0, 0), (−5, 0), (5, 0), and (−2,
0). Further,
the leading term of the polynomial is x^{4}, so the polynomial must have the same
endbehavior
as y = x^{4}, namely, it must fall from positive infinity, wiggle through its
xintercepts, then rise to positive infinity. The sketch with the appropriate
zeros and
end behavior follows.
Checking on the calculator.
41. Start with p(x) = −3x^{4} − 9x^{3} + 3x^{2} + 9x, then factor out the gcf (−3x in
this
case). Then, factor by grouping and complete the factorization with the
difference of
squares pattern.
p(x) = −3x[x^{3} + 3x^{2} − x − 3]
p(x) = −3x[x^{2}(x + 3) − 1(x + 3)]
p(x) = −3x(x^{2} − 1)(x + 3)
p(x) = −3x(x + 1)(x − 1)(x + 3)
Using the zero product property, the zeros are 0, −1, 1,
and −3. Hence, the graph of
the polynomial must intercept the xaxis at (0, 0), (−1, 0), (1, 0), and (−3,
0). Further,
the leading term of the polynomial is −3x^{4}, so the polynomial must have the same
endbehavior as y = −3x^{4}, namely, it must rise from negative infinity, wiggle
through
its xintercepts, then fall back to negative infinity . The sketch with the
appropriate
zeros and end behavior follows.
Checking on the calculator.
43. Start with p(x) = −x^{3}−x^{2}+20x, then factor out the gcf (−x in this case).
Then,
complete the factorization with the acmethod.
p(x) = −x[x^{2} + x − 20]
p(x) = −x(x + 5)(x − 4)
Using the zero product property, the zeros are 0, −5, and
4. Hence, the graph of the
polynomial must intercept the xaxis at (0, 0), (−5, 0), and (4, 0). Further,
the leading
term of the polynomial is −x^{3}, so the polynomial must have the same
endbehavior as
y = −x^{3}, namely, it must fall from positive infinity, wiggle through its
xintercepts,
then fall to negative infinity. The sketch with the appropriate zeros and end
behavior
follows.
Checking on the calculator.
45. Start with p(x) = 2x^{3} +3x^{2} −35x, then factor out the gcf (x in this case).
Then,
complete the factorization with the acmethod.
p(x) = x[2x^{2} + 3x − 35]
p(x) = x[2x^{2} − 7x + 10x − 35]
p(x) = x[x(2x − 7) + 5(2x − 7)]
p(x) = x(x + 5)(2x − 7)
Using the zero product property, the zeros are 0, −5, and
7/2. Hence, the graph of
the polynomial must intercept the xaxis at (0, 0), (−5, 0), and (7/2, 0).
Further, the
leading term of the polynomial is 2x^{3}, so the polynomial must have the same
endbehavior
as y = 2x^{3}, namely, it must rise from negative infinity, wiggle through its
xintercepts, then rise to positive infinity. The sketch with the appropriate
zeros and
end behavior follows.
Checking on the calculator.
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