Lesson test: Polynomials and their roots

1.[2p]
Which of these are polynomials?

Select all that apply

$3 x^{4} - x^{3} + 7 x - 2$

Correct

$x^{2} - 4$

Correct

$1 / x + 3$

The answer is: $3x^4 - x^3 + 7x - 2$, $x^2 - 4$

$2^{x}$

The answer is: $3x^4 - x^3 + 7x - 2$, $x^2 - 4$

$\sqrt{x} + 1$

The answer is: $3x^4 - x^3 + 7x - 2$, $x^2 - 4$

2.[2p]
What is the remainder when $x^{3} - 4 x^{2} + 2 x + 5$ is divided by $x - 3$ ?

CorrectNot quite: 2

3.[2p]
What does the factor theorem say?

$p (a) = 0$ if and only if $x - a$ is a factor of $p$

Correct

Every polynomial factors into linear pieces over the reals

The answer is: $p(a) = 0$ if and only if $x - a$ is a factor of $p$

A polynomial of degree $n$ has exactly $n$ real roots

The answer is: $p(a) = 0$ if and only if $x - a$ is a factor of $p$

A repeated factor makes the graph cross the axis twice

The answer is: $p(a) = 0$ if and only if $x - a$ is a factor of $p$

4.[3p]
$2$ is a root of $2 x^{3} + x^{2} - 13 x + 6$ . What is the smallest of its three roots?

CorrectNot quite: -3

5.[2p]
At a root of multiplicity two, what does the graph do?

Touches the axis and turns back without crossing

Correct

Crosses the axis steeply

The answer is: Touches the axis and turns back without crossing

Has a vertical asymptote

The answer is: Touches the axis and turns back without crossing

Crosses the axis twice at nearby points

The answer is: Touches the axis and turns back without crossing

6.[3p]
Give the four roots of $x^{4} - 5 x^{2} + 4$ in increasing order, separated by commas.

CorrectNot quite: -2, -1, 1, 2

7.[2p]
The fundamental theorem of algebra guarantees that every polynomial of degree $n$ has $n$ real roots.

True

The answer is: False

False

Correct

8.[2p]
Why is there no general formula in radicals for the fifth-degree equation?

Abel proved in 1824 that no such formula can exist

Correct

Nobody has managed to find one yet

The answer is: Abel proved in 1824 that no such formula can exist

Because fifth-degree equations have no roots

The answer is: Abel proved in 1824 that no such formula can exist

Because the rational root theorem fails above degree four

The answer is: Abel proved in 1824 that no such formula can exist

9.[3p]
A cubic has roots $- 1$ , $2$ and $5$ and passes through $(0, 20)$ . What is its leading coefficient?

CorrectNot quite: 2

1.[2p] Which of these are polynomials?

2.[2p] What is the remainder when x3-4x2+2x+5 is divided by x-3?

3.[2p] What does the factor theorem say?

4.[3p] 2 is a root of 2x3+x2-13x+6. What is the smallest of its three roots?

5.[2p] At a root of multiplicity two, what does the graph do?

6.[3p] Give the four roots of x4-5x2+4 in increasing order, separated by commas.

7.[2p] The fundamental theorem of algebra guarantees that every polynomial of degree n has n real roots.

8.[2p] Why is there no general formula in radicals for the fifth-degree equation?

9.[3p] A cubic has roots -1, 2 and 5 and passes through (0,20). What is its leading coefficient?

1.[2p]
Which of these are polynomials?

2.[2p]
What is the remainder when $x^{3} - 4 x^{2} + 2 x + 5$ is divided by $x - 3$ ?

3.[2p]
What does the factor theorem say?

4.[3p]
$2$ is a root of $2 x^{3} + x^{2} - 13 x + 6$ . What is the smallest of its three roots?

5.[2p]
At a root of multiplicity two, what does the graph do?

6.[3p]
Give the four roots of $x^{4} - 5 x^{2} + 4$ in increasing order, separated by commas.

7.[2p]
The fundamental theorem of algebra guarantees that every polynomial of degree $n$ has $n$ real roots.

8.[2p]
Why is there no general formula in radicals for the fifth-degree equation?

9.[3p]
A cubic has roots $- 1$ , $2$ and $5$ and passes through $(0, 20)$ . What is its leading coefficient?