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Solutions to graded problems
Section 3.2
2. Using long division:
So P(x) = (3x2)(x^{2}4)12.
10. Write P(x) = 2x^{3}10x^{2}28x+60.
Then we are told that x =3 is a solution of the
equation P (x) = 0, which means that (x + 3) must be a factor of P(x), by
the factor
theorem. We use long division to factor P :
So now the original equation is equivalent to
2(x + 3)(x^{2}8x + 10) = 0
(I pulled out a 2 for convenience.) To find the remaining two solutions of this
equation
(apart from x =3), we can use the quadratic formula on the second bracket:
(x^{2}8x + 10) = 0⇔⇔
So the equation has solutions x =3, x = 4and
x = 4 +.
16. We're told to use a graph to solve the equation x^{3}3x^{2}4x+12
= 0. This corresponds
to looking for x intercepts of the graph y=x^{3}3x^{2}4x+12,
sketched below:
From here, we can immediately read o that the xintercepts are at (2,0), (2,0)
and (3,0). By the factor theorem , this means we can write x^{3}3x^{2}4x+12=
(x3)(x2)(x + 2). So the original equation has solutions x =2, x = 2 and x =
3.
28. How do we solve a cubic equation such as this one?
First, collect all terms to one side:
x^{3} + 10x^{2} + 14x60 = 0
From this point, our strategy is to inspect the graph and try to find at least one
x 
intercept with rational coordinates (this is what we will do in general, for a
polynomial
equation with degree≥3). The graph is sketched below:
On your calculators, you should use the calculate function to find the
xintercepts
(carefully!) I have drawn a zoomed in graph (below) to show what happens near
x =6.
From your calculator, you should find that the xintercepts are at x =6, x≈
5.741657...and x≈1.741657...Obviously, the latter two are not exact. To find
them exactly, we can factor our polynomial, using the information that x =6 is
an
exact zero:
So x^{3}3x^{2}4x+12= 0 , (x+6)(x^{2} +4x10) = 0 )⇒x
=6 or x =,
So the final, exact solutions are x =6, x =2and
x =2 +
60. The information about the zeros gives that P(x) =
a(x1)^{2}(x + 1)^{2}. a is a stretching
factor that will be determined by the other piece of information: P(2) = 4 , 4 =
a(1)^{2}(3)^{2} = 9a⇔a =.So
P(x) =.
62. The graph shows that the polynomial P has zeros at x
=5, x = 2 and x = 5 (all
multiplicity one ), so we immediately have that P(x) = a(x5)(x2)(x + 5) (by the
factor theorem). To find the stretch factor a, we use one other point on the
graph: in
this case, the yintercept (0,50): P(0) = 50⇔50 = a(5)(2)(5) = 50a⇔a =
1. So P(x) = (x5)(x2)(x + 5).
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