# 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 x-axis 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 x-axis 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 x-axis 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 end-behavior

as y = 5x^{3}, namely, it must rise from negative infinity, wiggle through its

x-intercepts, 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 x-axis 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
end-behavior

as y = 4x^{3}, namely, it must rise from negative infinity, wiggle through its

x-intercepts, 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 x-axis 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
end-behavior

as y = x^{4}, namely, it must fall from positive infinity, wiggle through its

x-intercepts, 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 x-axis 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

end-behavior as y = −3x^{4}, namely, it must rise from negative infinity, wiggle
through

its x-intercepts, 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 ac- method .

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 x-axis 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
end-behavior as

y = −x^{3}, namely, it must fall from positive infinity, wiggle through its
x-intercepts,

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 ac-method.

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 x-axis 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
end-behavior

as y = 2x^{3}, namely, it must rise from negative infinity, wiggle through its

x-intercepts, then rise to positive infinity. The sketch with the appropriate
zeros and

end behavior follows.

Checking on the calculator.

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