Polynomials

The polynomial equation in standard form that has the roots {-2, -1, 1, 3} is

x^4 - x^3 - 7x^2 + x + 6

The roots of the polynomial equation x (x + 3) (x - 3) (x + 2) (x2+ 1) = 0 are

{-3, -1/2, 3}

The roots of the x5 + 2x4 = 4x3 +8x2 are

{2, 0 with multiplicity of 2, -2 with multiplicity of 2}

What is the remainder when 6x2 + 17x + 5 is divided by 3x + 1? (use Remainder Theorem)

0

How many positive roots are there in the polynomial equation x3+ 2x2- x - 2 = 0?

1

How many roots are there in the polynomial equation x3 + 2x2- 3x + 4 = 0?

3

How many factors are there in a quartic polynomial?

4

A quadratic equation has two factors. (True or False)

True

How many times will the graph of P(x) = (x + 6) (2x - 1) (3x + 4) intersect the x-axis? a.) 3 b.) 4 c.) 2 d.) None of the Choices e.) 1

a

What type of polynomial when the degree is 4 with 3 terms? a.) None of the Choices b.) Quartic Trinomial c.) Cubic Polynomial d.) Quintic Polynomial e.) Quadratic Trinomial

b

The quotient when x2 - 2x - 35 is divided by x - 7 is

x + 5

What value of k makes (x + 3) a factor of x3+ 2x2- kx + 3?

2

A polynomial equation with degree of 1 is called linear equation. (True or False)

True

The negative roots in the polynomial equation x3 + 2x2- x - 2 = 0 are

{2, -1, 1}

The leading coefficient of 4x3+ 2x2 - 3x5 + 1 is

-3

How many roots are there in the polynomial equation x (x + 3) (x - 3) (x + 2) (x2+ 1) = 0?

6

How many roots are there in the polynomial equation x(x + 3) (x - 3) (x + 2) (x2+ 1) = 0?

6

At which point the graph of x3 - x2 - 6x will not cross the x-axis? a.) (- 3, 0) b.) (- 2, 0) c.) (3, 0) d.) None of the Choices e.) (0, 0)

a

Given the polynomial m (m - 4)(m +5)(3m + 1) = 0, which is its complete set of the roots? a.) {-5, -1/3, 0, 4} b.) { -5, 1, 4} c.) {-5, 0, 1/3, 4} d.) None of the Choices e.) {-5, 3, 4}

a

Given: P(x) = -2 (x - 1) (x + 5)3(x - 4)3 I. Its graph is decreasing on both sides. II. The roots of the polynomial function are - 1, 5 with a multiplicity of 3 and - 4 with a multiplicity of 3 a.) statements I and II are FALSE b.) only statement II is TRUE c.) only statement I is TRUE d.) statements I and II are TRUE e.) None of the Choices

a

Given: m3 - 11m2 + 24m = 0 I. - 3 is a root. II. The polynomial has three positive roots. a.) statements I and II are FALSE b.) None of the Choices c.) statements I and II are TRUE d.) only statement II is TRUE e.) only statement I is TRUE

a

I. The graph of a polynomial function with a degree of 2 and whose roots are rational means it intersects the x-axis twice. II. The graph of an odd function with a positive leading coefficient rises to the left and falls to the right. a.) only statement I is TRUE b.) only statement II is TRUE c.) statements I and II are TRUE d.) statements I and II are FALSE e.) None of the Choices

a

What makes P(x)= x^3 - 3 / x not a polynomial? a.) Its denominator has a variable. b.) Its degree is a fraction. c.) None of the Choices d.) Its degree is 3. e.) Its numerator has a variable

a

Which zero has an odd multiplicity in P(x) = x2(x - 3)4(x + 3)3? a.) -3 b.) -1 c.) 3 d.) None of the Choices e.) 0

a

If one of the roots of the polynomial equation 2x3 + 5x2 - 9x - 18 = 0 is 2, then what are the other roots? a.) - 3/2 and 3 b.) - 3/2 and - 3 c.) 3/2 and - 3 d.) 3/2 and 3 e.) None of the Choices

b

What is the end behavior of the polynomial function f(x)= - 4x2 - 2x + 1 a.) rises to the left and rises to the right b.) falls to the left and falls to the right c.) rises to the left and falls to the right d.) None of the Choices e.) falls to the left and rise to the right

b

Refer to the graph below: I. The graph is for a quartic polynomial. II. The leading coefficient is negative. (wavy path up to the right, down to the left) a.) only statement II is TRUE b.) only statement I is TRUE c.) statements I and II are TRUE d.) statements I and II are FALSE e.) None of the Choices

d

Which BEST describes the graph of f(x) = x5 + 8x4 + 4x - 3? a.) It rises to the left and right. b.) It falls to the left and right. c.) It rises to the left and falls to the right. d.) It rises to the right and falls to the left. e.) None of the Choices

d

Which best describes the graph at the right? (W Graph) a.) It has a negative first coefficient and even degree b.) None of the Choices c.) It has a positive first coefficient and odd degree d.) It has a positive first coefficient and even degree e.) It has a negative first coefficient and odd degree

d

Which divisor will give a zero remainder when used as a divisor for x5 + 3x4 - 5x3 - 15x2 + 4x + 12 but a non-zero remainder less than 20 when used as a divisor for x3 + 5x2 + 8x + 4 ? a.) x + 2 b.) x - 2 c.) x + 1 d.) x - 1 e.) None of the Choices

d

Which is a polynomial equation in standard form? a.) x^2 - 4 b.) (x + 2) (x - 2) c.) None of the Choices d.) x^2 - 4 = 0 e.) P(x)= x^2 - 4

d

Which value of k makes (x - 1) a factor of x4 + 3x3 + 2kx2 + 2x + 6? a.) 6 b.) 3 c.) None of the Choices d.) -6 e.) -3

d

Referring to the graph below, which statement is FALSE? (W graph) a.) None of the Choices b.) The degree is even. c.) The first coefficient is positive. d.) The degree equals the number of turning points e.) It has a minimum value.

e

What value of k gives a remainder of 7 when (2x3 + 5x2 - kx - 5) is divided by (x + 3)? a.) -5 b.) 5 c.) 2 d.) None of the Choices e.) 7

e

Which BEST describes the graph of f(x) = - x3 + 4x2 + 4x? a.) It is decreasing. b.) It is increasing. c.) None of the Choices d.) It rises to the right and falls to the left. e.) It rises to the left and falls to the right.

e

Which is NOT a root of y (y + 3) (y + 3) (x - 1) (2x - 1) = 0? I. 0 II. −3 III. −1 IV. ½ a.) None of the Choices b.) II only c.) III and IV only d.) I and II only e.) III only

e

Which is a polynomial? a.) 3/x b.) 3√x c.) None of the Choices d.) x^-3 e.) x√3

e

Which is the answer if 3x - 4 divides 6x^3 + x^2 - 12x + 7? a.) None of the Choices b.) 2x^2 + 3x c.) (2x^2 + 3x) (2x - 4) + 7 d.) 2x^2 + 3x + 7 e.) 2x^2 + 3x + 7 / 3x-4

e

Which polynomial equation has 4 roots? a.) x^4 - 4x^5 - 7 = 0 b.) x^3 - 4x - 2 = 0 c.) x^3 - 4x^4 - 6x^6 = 0 d.) None of the Choices e.) 2x + 8x^3 - 4x^4 - 11 = 0

e

The roots of the polynomial equation 2x4 + 8x2 = 0 are

{0 with multiplicity of 2, 2i, -2i}

If the remainder is 1, then the divisor is a factor of the dividend. (True or False)

False

What should be the value of n for P(x) = xn be considered a polynomial function? a.) a non-negative integer b.) an integer c.) any real number except zero d.) None of the Choices e.) any real number

a

Which is TRUE about P(x) = (x + 6) (3x + 4) (x - 3)? a.) It is an odd positive function. b.) None of the Choices c.) It is an even negative function. d.) It is an odd negative function. e.) It is an even positive function.

a

Which is/are TRUE about x4 - 5x2 + 4 = 0? I. One of its roots is - 2. II. One of its factors is (x - 1). III. One of its roots has a multiplicity. a.) I and II only b.) None of the Choices c.) II only d.) I only e.) II and III only

a

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. f(x)= -x2 (x+ 9) (x2 - 1) a.) None of the Choices b.) 0, touches the x-axis and turns around; -9, crosses the x-axis; -1, crosses the x-axis; 1, crosses the x-axis c.) 0, touches the x-axis and turns around; -9, crosses the x-axis; 1, touches the x-axis and turns around d.) 0, crossesthe x-axis; -9, crosses the x-axis; -1, crosses the x-axis; 1,crosses the x-axis e.) 0, touches the x-axis and turns around; 9, crosses the x-axis; -1, touches the x-axis and turns around; 1, touches the x-axis and turns around

b

How many factors does a quintic polynomial in x have? a.) 3 b.) 5 c.) 4 d.) 2 e.) None of the Choices

b

How many positive real roots does x4 − x3 − 11x2 + 9x + 18 = 0 have? a.) None of the Choices b.) 2 c.) 1 d.) 3 e.) 0

b

If a function has zeros at -2, 0, and 4, then which may be its equivalent polynomial function? a.) None of the Choices b.) f(x) = x(x + 2)(x - 4) c.) f(x) = -x (x + 2)(x + 4) d.) f(x) = x(x - 2)(x + 4) e.) f(x) = -x(x - 2)(x - 4)

b

What polynomial equation in factored form has the roots {-3, 1, 2}? a.) (x-3)(x-1)(x-2) = 0 b.) (x+3)(x-1)(x-2) = 0 c.) (x+3)(x+1)(x+2) = 0 d.) (x-3)(x-1)(x-2) = 0 e.) None of the Choices

b

Which cubic polynomial equation has roots −2, 2 and 4 ? a.) x^3 + 4x^2 - 4x + 16 = 0 b.) x^3 - 4x^2 - x +16 = 0 c.) x^3 - 4x^2 - 4x +16 = 0 d.) 10x^3 - x^2 -x +16 = 0 e.) None of the Choices

b

Which is NOT a zero of P(x) = x5 - 5x3 + 4x? a.) -1 b.) 4 c.) None of the Choices d.) -2 e.) 0

b

Which is TRUE about the graph of x3 + 2x2 - x - 2? a.) It only touches the x-axis three times. b.) It is increasing to the right and decreasing to the left. c.) It crosses the x-axis twice. d.) It is increasing to the left and to the right. e.) None of the Choices

b

Which is TRUE about the polynomial whose factors are (x - 1), (x - 2)2 and (x - 3)? a.) Its degree is 3 b.) It has a root with multiplicity. c.) None of the Choices d.) Its leading coefficient is negative. e.) Its constant term is negative

b

Which is TRUE about the roots of the polynomial P(x) = 2x4 + 9x3 - 34x2 + 9x + 14? a.) All of them are integers. b.) Their sum is negative. c.) None of the Choices d.) Their sum is positive e.) All of them are natural numbers

b

Which is the complete factored form of x4 - x3 - x2 + x = 0? a.) None of the Choices b.) (x) (x - 1) (x - 1) (x + 1) = 0 c.) (x) (x - 1) (x + 1) (x + 1) = 0 d.) (x) (x + 1) (x - 1) = 0 e.) (x) (x + 1) (x^2 +1) = 0

b

Which is the leading coefficient of the polynomial function P(x) = - 5x3 + 1 + 3x2 - 4x4 ? a.) 5 b.) -4 c.) None of the Choices d.) 4 e.) -5

b

Which polynomial has a remainder when divided by x - 3? a.) x^4 - 81 b.) x^4 + 81 c.) -x^3 + 9x^2 - 27x +27 d.) x^3 - 9x^2 +27x -27 e.) None of the Choices

b

Which value of m will make the graph of the function P(x) = 2xm (x - 1)2 (x + 2)3 decreasing to the left and increasing to the right a.) None of the Choices b.) 2 c.) -2 d.) 1 e.) -1

b

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. f(x) = x4- 49x2 a.) 0, touches the x-axis and turns around; 49, touches the x-axis and turns around b.) 0, crosses the x-axis; 7, crosses the x-axis; -7, crosses the x-axis c.) 0, touches the x-axis and turns around; 7, crosses the x-axis; -7, crosses the x-axis d.) 0, touches the x-axis and turns around; 49,crosses the x-axis e.) None of the Choices

c

Is (x +1) a factor of (x2 - 2x + 1)? a.) No, because the factors of x^2 - 2x + 1 are (x + 1) and (x - 2). b.) None of the Choices c.) No, because when x^2 - 2x + 1 is divided by (x +1), the remainder is 4. d.) Yes, because when (x + 1) is multiplied by (x + 1), the product is x^2 - 2x + 1. e.) Yes, because when (x + 1) is multiplied by (x - 1), the product is x^2 - 2x + 1.

c

Which is FALSE about the graph of P(x) = -x4 - 6x3 - 9x2 + 4x + 12? a.) The degree of the function is 4. b.) The graph decreases to the left and right. c.) The graph of the function may have a maximum of 4 turning points. d.) None of the Choices e.) The graph is continuous

c

Which is FALSE about x^3 - 3x^5 + 4 - x? a.) It's an example of quintic polynomials. b.) None of the Choices c.) If it is used as a dividend, then there is one place holder. d.) Its leading coefficient is a negative integer. e.) If it will be divided by x + 1, then the remainder to be computed is 7.

c

Which is FALSE in dividing (x2 - 16x5 + 4x3 - 3) by (x - 2)? a.) There will be placeholders if long division will be used. b.) None of the Choices c.) The terms should be arranged in decreasing degree before using the Remainder Theorem. d.) There is a remainder. e.) Synthetic division can be used.

c

Which polynomial function will have the least value of the remainder when divided by x - 2? a.) P(x)= 3x^3 - x^2 + 2x +3 b.) None of the Choices c.) P(x)= 3x^3 - 2x^2 + x -1 d.) P(x)= 2x^3 + 2x^2 - 3x + 4 e.) P(x)= 2x^3 + 3x^2 - x - 4

c

Find the zeros of f(x) = (x + 2)6 (x + 3)4 and state the multiplicity. a.) 6, multiplicity -2; 4, multiplicity -3 b.) 6, multiplicity -2; -3, multiplicity 4 c.) None of the Choices d.) -2, multiplicity 6; -3, multiplicity 4 e.) -2, multiplicity 6; 4, multiplicity -3

d

Given: P(x) = (x + 3) (x - 3) (x + 2)2 I. {± 3} has both 0 multiplicities. II. x = 2 with the multiplicity of 2. a.) only statement II is TRUE b.) statements I and II are TRUE c.) only statement I is TRUE d.) statements I and II are FALSE e.) None of the Choices

d

Given: f(x) = k (x - s) (x - t)2 The given is a cubic function. Which statement is TRUE? a.) It has two zeros b.) None of the Choices c.) It has no turning point. d.) It has three zeros e.) It has three turning points

d

Given: f(x) = k (x - s) (x - t)2 What do k indicates? a.) the existence of a local maximum b.) the number of turning points c.) the direction of the graph d.) the behavior of the endpoints e.) None of the Choices

d