Topic 2: Polynomials

Factor out the greatest common factor first, then look for a difference of squares or grouping. The remainder when p(x) is divided by x minus a is p(a), so p(a) = 0 means x minus a is a factor.

A lesson for this topic

Part I style: multiple choice

Question 1Part I style

Which is the complete factorization of

2x^{3}

{}- 3x^{2}

{}- 8x

{}+ 12

Question 2Part I style

Which expression is equivalent to

27x^{3} + 8

Question 3Part I style

Which is

16x^{4} - 1

factored completely?

Question 4Part I style

Which expression is equivalent to

(x^{2} - 4)^{2}

{}- (x - 2)^{2}

Question 5Part I style

What are the zeros of

f(x)

= 6x^{2} + x - 15

Question 6Part I style

Which expression is a factor of

x^{3} - 64

Question 7Part I style

Which is

x^{6} - 64

factored completely over the integers?

Question 8Part I style

What are all the real zeros of

f(x)

= x^{4} - 5x^{2} - 36

Question 9Part I style

What is the remainder when

p(x)

= 2x^{3}

{}- 5x^{2}

{}+ 4x

{}+ 5

is divided by

x - 3

Question 10Part I style

When the polynomial

p(x)

is divided by

x + 4

, the remainder is 0. Which statement must be true?

Question 11Part I style

When

x^{3} + kx^{2} - 3x + 6

is divided by

x - 2

, the remainder is 12. What is the value of

k

Question 12Part I style

The binomial

x + 3

is a factor of

p(x)

= x^{3}

{}+ 2x^{2}

{}- 5x

{}- 6

. What are all the zeros of

p(x)

Question 13Part I style

Which function has zeros

-2

\frac{1}{3}

and

4

Question 14Part I style

Values of p(x): at x = −2, p(x) = −28; at x = −1, p(x) = 0; at x = 0, p(x) = 6; at x = 1, p(x) = 2; at x = 2, p(x) = 0.
$x$	$p(x)$
−2	−28
−1	0
0	6
1	2
2	0

The table shows some values of a cubic polynomial function

p(x)

whose leading coefficient is 2. What is the third zero of

p(x)

Question 15Part I style

For

p(x)

= x^{3}

{}+ ax^{2}

{}+ bx

{}- 12

, the binomial

x - 2

is a factor, and when

p(x)

is divided by

x + 1

the remainder is

-6

. What are the values of

a

and

b

Question 16Part I style

The graph of the polynomial function

y

= p(x)

is shown. What is the remainder when

p(x)

is divided by

x - 2

Question 17Part I style

The graph of

f(x)

= x(2x - 3)^{2}(x + 5)

meets the

x

-axis at three points. At which value of

x

does the graph touch the

x

-axis without crossing it?

Question 18Part I style

A small company models its weekly profit,

P(x)

, in hundreds of dollars, from selling

x

hundred jars of jam as

P(x)

= -x^{3}

{}+ 12x^{2}

{}- 20x

, for

0

\leq x

\leq 12

. For which values of

x

does the company make a profit?

Question 19Part I style

Which statement describes the end behavior of

f(x)

= -2x^{5} + 3x^{2} - 7

Question 20Part I style

Which equation could represent the graph shown?

Question 21Part I style

Which statement about the graph of

f(x)

= x(x - 3)^{2}(x + 1)^{3}

is true?

Question 22Part I style

A polynomial function has zeros

-4

0

and

3

only. As

x \to -\infty

f(x) \to \infty

, and as

x \to \infty

f(x) \to -\infty

. Which could be the function?

Question 23Part I style

The function

f(x)

= x^{3}

{}- 6x^{2}

{}+ 9x

{}+ 1

has a relative maximum at

(1, 5)

and a relative minimum at

(3, 1)

. For which values of

x

f(x)

decreasing?

Constructed response: Parts II, III and IV

Factoring a cubic completelyPart II style · 2 credits

The expression

x^{3}

{}- 3x^{2}

{}- 9x

{}+ 27

has four terms.

(a) Factor

x^{3}

{}- 3x^{2}

{}- 9x

{}+ 27

completely. Show your work. [2]

Show your work on paper; type the final answer here. Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

Testing a factorPart II style · 2 credits

Let

p(x)

= 2x^{3}

{}+ 5x^{2}

{}- 4x

{}- 3

(a) Find the remainder when

p(x)

is divided by

x + 3

. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

(b) Explain whether

x + 3

is a factor of

p(x)

. [1]

0 words · a sentence

A factor with a missing coefficientPart III style · 4 credits

The polynomial

p(x)

= 2x^{3}

{}+ x^{2}

{}+ kx

{}+ 6

, where

k

is a constant, has

x - 2

as a factor.

(a) Find the value of

k

. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

(b) Write

p(x)

as a product of linear factors. Show your work. [2]

Show your work on paper; type the final answer here. Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

p(x)

. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

Designing a gift boxPart IV style · 6 credits

A craft company makes gift boxes shaped like rectangular prisms. For a box of width

x

inches, the width and the height add to 12 inches, and the length is 4 inches more than the width. The volume of the box, in cubic inches, is

V(x)

= x(x + 4)(12 - x)

(a) Write

V(x)

as a polynomial in standard form. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

(b) State all the zeros of

V(x)

. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

0 words · a sentence

(d) Find, to the nearest tenth of an inch, the width that gives the greatest volume. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

(e) Find the greatest volume, to the nearest cubic inch. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

(f) A box of width 4 inches has a volume of 256 cubic inches. Find, to the nearest tenth of an inch, the other width that gives a volume of 256 cubic inches. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.