A.APR.B.3.a: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Example Problems

Factor and solve:

16x^3 - 48x^2 - 64x = 0

Consider the polynomial function

P(x) = x^4 - 3x^3 + ax^2 - 6x + 14

,
where

a

is an unknown real number.

If

(x - 2)

is a factor of this polynomial, what is the value of

a

?

Suppose

f

is a

\text{quadratic function}

given by the equation

f(x) = ax^2 + bx + c

where

a, b, c

are real numbers and

a

is non-zero.

a. Explain why

f

can have at most two

\text{roots}

; that is, explain why there can be at most two distinct real numbers

r_1, r_2

so that

f(r_1) = f(r_2) = 0

.

Give an example to show that it is possible for

f

to have:
b. Zero roots
c. One root
d. Two real

\text{roots}

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