POLYNOMIAL FUNCTIONS

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Test your understanding of polynomial functions, including identifying degrees, leading coefficients, end behavior, factoring, graph interpretation, and solving polynomial equations.

1 / 55

When dividing

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

$x - 1$

, the quotient is:

x^{2} + x + 2

x^{2} - x - 2

x^{2} - x + 2

x^{2} + x - 2

2 / 55

The remainder when $2 x^{3} - 5 x^{2} + 4 x - 1$ is divided by $x - 2$ is:

-5

-3

Substitute $x = 2$ into the polynomial: $2 (8) - 5 (4) + 4 (2) - 1 = 3$ .

3 / 55

$x^{4} - 3 x^{3} + 2 x^{2} - x + 5$

is divided by

$x^{2} - 1$

, the remainder is:

- 2 x + 3

2 x + 3

3 x + 2

No remainder

4 / 55

Synthetic division is used to divide a polynomial by:

Any polynomial

Linear divisors of the form

x - c

Only quadratic polynomials

Only monic polynomials

5 / 55

The quotient when

$x^{3} - 6 x^{2} + 11 x - 6$

is divided by

$x - 3$

is:

x^{2} + 3 x + 2

x^{2} + 3 x - 2

x^{2} - 3 x - 2

x^{2} - 3 x + 2

6 / 55

When dividing $3 x^{4} - 4 x^{3} + 2 x^{2} - 5 x + 1$ by $x + 1$ , the remainder is:

-15

-7

Substitute $x = - 1$ : $3 (1) - 4 (- 1) + 2 (1) - 5 (- 1) + 1 = 15$ .

7 / 55

The degree of the quotient when a 4th-degree polynomial is divided by a 2nd-degree polynomial is:

8 / 55

$x^{3} + 2 x^{2} - 5 x - 6$

is divided by

$x + 3$

, the quotient is:

x^{2} - x - 2

x^{2} + x - 2

x^{2} - x + 2

x^{2} + x + 2

9 / 55

The remainder when $x^{5} - 1$ is divided by $x - 1$ is:

-1

$x - 1$ is a factor, so remainder is 0.

10 / 55

The division algorithm states that for polynomials

$P (x)$

and

$D (x) \neq 0$

, there exist unique polynomials

$Q (x)$

and

$R (x)$

such that:

P (x) = D (x) \cdot Q (x) + R (x)

P (x) = D (x) + Q (x) \cdot R (x)

P (x) = Q (x) / D (x) + R (x)

P (x) = D (x) - Q (x) \cdot R (x)

11 / 55

The Remainder Theorem states that the remainder when

$P (x)$

is divided by

$x - c$

is:

P (- c)

P (0)

P (c)

P (1)

12 / 55

If $P (x) = 2 x^{3} - 3 x^{2} + 4 x - 1$ , then $P (1)$ equals:

-2

$P (1) = 2 (1) - 3 (1) + 4 (1) - 1 = 2$ .

13 / 55

The remainder when $x^{4} - 2 x^{3} + x^{2} - 3 x + 5$ is divided by $x - 2$ is:

-7

-11

Substitute $x = 2$ : $16 - 16 + 4 - 6 + 5 = 7$ .

14 / 55

$P (x)$

leaves a remainder of 5 when divided by

$x - 3$

, then:

P (- 3) = 5

P (5) = 3

P (0) = 5

P (3) = 5

15 / 55

The remainder when $3 x^{3} - 4 x^{2} + 5 x - 2$ is divided by $x + 1$ is:

-14

-10

Substitute $x = - 1$ : $- 3 - 4 - 5 - 2 = - 14$ .

16 / 55

If $P (x) = x^{3} + k x^{2} - 4 x + 3$ and $P (2) = 7$ , then $k$ is:

-1

-2

$8 + 4 k - 8 + 3 = 7 \Rightarrow k = 1$ .

17 / 55

The remainder when $x^{5} + 2 x^{3} - x^{2} + 4$ is divided by $x - 1$ is:

-6

-4

Substitute $x = 1$ : $1 + 2 - 1 + 4 = 6$ .

18 / 55

$P (x)$

is divided by

$x - c$

and the remainder is 0, then:

x - c

is a factor of

P (x)

P (x)

has no real roots

P (c) = 1

P (x)

is of degree 1

19 / 55

The remainder when

$4 x^{3} - 3 x^{2} + 2 x - 1$

is divided by

$x - \frac{1}{2}$

is:

- \frac{1}{2}

\frac{1}{2}

Substitute

$x = \frac{1}{2}$

$4 (\frac{1}{8}) - 3 (\frac{1}{4}) + 2 (\frac{1}{2}) - 1 = - \frac{1}{2}$

20 / 55

If $P (x) = x^{3} - 4 x^{2} + 5 x - 2$ , then $P (2)$ is:

-1

$P (2) = 8 - 16 + 10 - 2 = 0$ .

21 / 55

The Factor Theorem states that

$x - c$

is a factor of

$P (x)$

if and only if:

P (c) = 0

P (c) = 1

P (0) = c

P (1) = c

22 / 55

Which of the following is a factor of

$x^{3} - 3 x^{2} - 4 x + 12$

x + 1

x - 1

x - 2

x + 3

$P (2) = 8 - 12 - 8 + 12 = 0$

23 / 55

$x + 1$

is a factor of

$P (x) = x^{3} + 2 x^{2} - x - 2$

, then another factor is:

x + 2

x - 1

x - 2

x + 3

Factor as

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

24 / 55

The polynomial

$P (x) = x^{3} - 6 x^{2} + 11 x - 6$

has factors:

(x - 1) (x - 2) (x - 3)

(x + 1) (x + 2) (x + 3)

(x - 1) (x + 2) (x - 3)

(x + 1) (x - 2) (x + 3)

25 / 55

If $x - 3$ is a factor of $x^{3} - k x^{2} + 5 x - 6$ , then $k$ is:

-4

-6

$P (3) = 27 - 9 k + 15 - 6 = 0 \Rightarrow k = 4$ .

26 / 55

The number of distinct real roots of $P (x) = x^{3} - 3 x^{2} + 2 x$ is:

Factor as $x (x - 1) (x - 2)$ ; roots at $x = 0, 1, 2$ .

27 / 55

$P (x) = x^{4} - 5 x^{2} + 4$

, then one of its factors is:

x - 1

x + 3

x - 4

x + 5

$P (1) = 1 - 5 + 4 = 0$

28 / 55

The polynomial

$P (x) = x^{3} + x^{2} - 4 x - 4$

has a factor:

x + 2

x - 3

x + 4

x - 5

$P (- 2) = - 8 + 4 + 8 - 4 = 0$

29 / 55

$x - \sqrt{2}$

is a factor of

$P (x) = x^{2} - 2$

, then another factor is:

x + \sqrt{2}

x - 2

x + 2

x - \sqrt{3}

$x^{2} - 2 = (x - \sqrt{2}) (x + \sqrt{2})$

30 / 55

The polynomial

$P (x) = x^{3} - 2 x^{2} - x + 2$

has roots:

1, - 1, 2

- 1, - 2, 3

0, 1, 2

1, 2, 3

Factor as

$(x - 1) (x + 1) (x - 2)$

31 / 55

The end behavior of $P (x) = - x^{3} + 2 x^{2} - x + 1$ is:

Falls left and right

Rises left, falls right

Rises left and right

Falls left, rises right

32 / 55

The number of turning points of $P (x) = x^{4} - 3 x^{2} + 2$ is at most:

A degree- $n$ polynomial has at most $n - 1$ turning points.

33 / 55

The graph of

$P (x) = (x - 1)^{2} (x + 2)$

crosses the x-axis at:

Both

x = 1

and

x = - 2

x = 1

(crosses) and

x = - 2

(touches)

Neither

x = - 2

(crosses) and

x = 1

(touches)

34 / 55

The y-intercept of

$P (x) = 2 x^{3} - 4 x^{2} + x - 5$

is:

(0, - 5)

(0, 5)

(0, - 2)

(0, 2)

Substitute

$x = 0$

35 / 55

The polynomial

$P (x) = x^{3} - 4 x$

has zeros at:

x = 0, \pm 1

x = 1, 2, 3

x = 0, \pm 2

x = - 1, - 2, - 3

Factor as

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

36 / 55

The graph of $P (x) = x^{2} (x - 3)$ resembles:

A parabola opening upwards

A cubic rising left and right

A cubic falling left, rising right

A line

37 / 55

The multiplicity of the root $x = 2$ in $P (x) = (x - 2)^{3} (x + 1)$ is:

The exponent of the factor $(x - 2)$ .

38 / 55

The graph of $P (x) = x^{4} - 1$ has symmetry about:

The y-axis

The x-axis

The origin

None

Even function: $P (- x) = P (x)$ .

39 / 55

The number of real roots of $P (x) = x^{3} + x^{2} + x + 1$ is:

Only $x = - 1$ is a real root; others are complex.

40 / 55

The graph of

$P (x) = x^{3} - 6 x^{2} + 11 x - 6$

has x-intercepts at:

x = - 1, - 2, - 3

x = 0, 1, 6

x = 1, 3, 6

x = 1, 2, 3

Factor as

$(x - 1) (x - 2) (x - 3)$

41 / 55

The solution to

$(x - 1) (x + 2) (x - 3) > 0$

is:

- 2 < x < 1

x > 3

x < - 2

1 < x < 3

x > 3

x < - 2

Test intervals defined by roots

$x = - 2, 1, 3$

42 / 55

The inequality

$x^{2} - 4 x + 3 \leq 0$

holds for:

x \leq 1

x \geq 3

1 \leq x \leq 3

All real numbers

Factor as

$(x - 1) (x - 3)$

; parabola opens upwards.

43 / 55

The solution to

$x^{3} - 4 x^{2} + x + 6 \geq 0$

includes:

x \in (- \infty, - 1] \cup [2, 3]

x \in [0, 1] \cup [4, \infty)

All real numbers

x \in [- 1, 2] \cup [3, \infty)

Factor as

$(x + 1) (x - 2) (x - 3)$

and test intervals.

44 / 55

The inequality

$x (x - 2) (x + 1) < 0$

is true for:

x < - 1

0 < x < 2

- 1 < x < 0

x > 2

x > 2

x < - 1

Test intervals defined by roots

$x = - 1, 0, 2$

45 / 55

The solution to

$x^{4} - 5 x^{2} + 4 \leq 0$

is:

x \leq - 2

x \geq 2

- 2 \leq x \leq - 1

1 \leq x \leq 2

- 1 \leq x \leq 1

All real numbers

Factor as

$(x^{2} - 1) (x^{2} - 4)$

and solve.

46 / 55

The inequality

$x^{3} + 2 x^{2} - x - 2 > 0$

holds for:

x > 1

- 2 < x < - 1

x < - 2

- 2 < x < - 1

x > 1

Factor as

$(x + 2) (x + 1) (x - 1)$

and test intervals.

47 / 55

The solution to

$(x + 3) (x - 1)^{2} \geq 0$

is:

x \leq - 3

x = 1

x \geq 1

x \geq - 3

x \leq 1

The squared term ensures non-negativity at

$x = 1$

48 / 55

The inequality

$x^{3} - 8 < 0$

is true for:

x < 2

x > 2

All real numbers

No solution

Factor as

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

; only

$x = 2$

is a real root.

49 / 55

The solution to

$x^{4} - 16 \geq 0$

is:

- 2 \leq x \leq 2

x \geq 0

All real numbers

x \leq - 2

x \geq 2

Factor as

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

50 / 55

The inequality

$(x - 1)^{3} (x + 2)^{2} \leq 0$

holds for:

x \leq 1

x \leq - 2

x = 1

x \geq - 2

x = - 2

x \geq 1

The squared term is always non-negative; the cubic term changes sign at

$x = 1$

51 / 55

The partial fraction decomposition of

$\frac{3 x + 2}{(x - 1) (x + 2)}$

is:

\frac{2}{x - 1} + \frac{1}{x + 2}

\frac{1}{x - 1} + \frac{2}{x + 2}

\frac{3}{x - 1} + \frac{2}{x + 2}

\frac{1}{x - 1} - \frac{1}{x + 2}

52 / 55

The partial fractions of

$\frac{x^{2} + 1}{x (x - 1)^{2}}$

include terms:

\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{(x - 1)^{2}}

\frac{A}{x} + \frac{B x + C}{(x - 1)^{2}}

\frac{A}{x} + \frac{B}{x - 1}

\frac{A}{x} + \frac{B}{(x - 1)^{2}}

53 / 55

The partial fraction decomposition of

$\frac{5 x - 1}{x^{2} - x - 2}$

is:

\frac{1}{x - 2} + \frac{4}{x + 1}

\frac{2}{x - 2} + \frac{3}{x + 1}

\frac{3}{x - 2} + \frac{2}{x + 1}

\frac{4}{x - 2} + \frac{1}{x + 1}

54 / 55

The partial fractions of

$\frac{2 x + 3}{(x + 1) (x^{2} + 1)}$

include:

\frac{A}{x + 1} + \frac{B}{x^{2} + 1}

\frac{A}{x + 1} + \frac{B}{x} + \frac{C}{x^{2} + 1}

\frac{A}{x + 1} + \frac{B x}{x^{2} + 1}

\frac{A}{x + 1} + \frac{B x + C}{x^{2} + 1}

55 / 55

The partial fraction decomposition of

$\frac{x^{2} + 2 x - 1}{x^{3} - x}$

is:

\frac{1}{x} + \frac{1}{x - 1} + \frac{1}{x + 1}

\frac{1}{x} + \frac{2}{x - 1} + \frac{- 1}{x + 1}

\frac{2}{x} + \frac{1}{x - 1} - \frac{1}{x + 1}

\frac{- 1}{x} + \frac{2}{x - 1} + \frac{1}{x + 1}

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