Question 1

Find all zeroes (real and complex) of the polynomial. p(x) = 4x³ - 5x² - 23x + 6

Accepted Answer

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Question 2

Gather information on the polynomial using the rational roots theorem, testing for 1 and -1, Descartes's rule of signs, and the upper and lower bounds property. Respond explicitly to each. p(x) = x⁴ - 3x³ + 9x - 27

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Question 3

State the degree and end behavior of the function. f(x) = (x - 1)⁶(x + 3)(x - 3)³

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Question 4

State the end behavior and y-intercept of the function. Do not graph. f(x) = x⁵ + 5x⁴ - 5x²- 3x + 5

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Question 5

Use the following to answer questions :  (Gridlines are spaced one unit apart.)
-Find the minimum possible degree of the function and write it in factored form. Assume all zeroes are real.

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Question 6

Use the following to answer questions :  (Gridlines are spaced one unit apart.)
-State whether the degree of the function is even or odd.

Accepted Answer

To determine if a function is odd or even, we look at how it behaves under the operation of reflection across the y-axis. If f(-x) = f(x), then the function is even. If f(-x) = -f(x), then the function is odd. In this case, we can see that the graph of the function is symmetric with respect to the y-axis, which means f(-x) = f(x). Therefore, the function is even.

Question 7

Use the following to answer questions :  (Gridlines are spaced one unit apart.)
-Use the graph to estimate the zeroes of the function, then state whether their multiplicity is even or odd.

Accepted Answer

From the graph, we can see that the function intersects the x-axis at approximately -1, 1, and 2. At -1, the curve touches the x-axis but does not cross it, so it has a root of multiplicity 2 (even). At 1 and 2, the curve crosses the x-axis, so they have roots of multiplicity 1 (odd). Therefore, the best choice is option A: -1, even; 2 odd.

Question 8

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial. f(x) = x³ - 3x - 2

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Question 9

For the complex polynomial below, one of its zeroes is z = 5i. Use the given zero to help find all zeroes of the polynomial, then write the polynomial in completely factored form. Hint: synthetic division and the quadratic formula can be applied to all polynomials, even those with complex coefficients. C(z) = z³ + (1 - 5i)z² + (-6 - 5i)z + 30i.

Accepted Answer

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Question 10

State the degree, end behavior, and y-intercept of the function. f(x) = -(x - 5)(x - 1)⁸(x - 2)²

Accepted Answer

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Question 11

State the end behavior of the function. f(x) = 5x⁴ + 2x³ - 5x²- 5x - 3

Accepted Answer

The leading term of the function $f(x) = 5x^{14} + 2x^{13} - 5x^{12} - 5x - 3$ is $5x^{14}$. Since the coefficient of the leading term (5) is positive and the degree (14) is even, both ends of the graph will go up as $x$ approaches infinity and negative infinity.

Question 12

Use the following to answer questions :
p(x) = xS1U1P14S1S1P0 - 5xS1U1P13S1S1P0 + 5xS1U1P12S1S1P0 + 5x - 6
-Use the information gathered, testing for 1 and -1, synthetic division, and the factor theorem to factor p completely.

Accepted Answer

Using synthetic division with 1 and -1 shows that both are roots of the polynomial, meaning it is divisible by (x-1) and (x+1). Then, using the factor theorem, we can see that (x+2) and (x+3) are also roots of the polynomial, making the factorization (x+1)(x-1)(x-2)(x-3). Therefore, the best choice is D.

Question 13

Find all zeroes (real and complex) of the polynomial. p(x) = 6x³ + 29x² - 149x + 24

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Question 14

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial. f(x) = x³ - 4x²+ x + 6

Accepted Answer

According to the Guidelines for Graphing Polynomial Functions, we should start by finding the zeros of the function. We can do this by setting the polynomial equal to zero and solving for x. 
x^13 - 4x^12 + x + 6 = 0 
Using a graphing calculator or software, we can find that the zeros are approximately -2.474, -1.635, -0.750, 0.128, and 1.282.

Next, we can determine the end behavior of the function by looking at the degree and sign of the leading coefficient. In this case, the leading coefficient is positive and the degree is odd, so the end behavior goes up on the left and down on the right.

We can also find the y-intercept by setting x equal to zero. 
f(0) = 6

Using this information, we can sketch the graph of the function. Choice C is the best option as it matches the description of the graph.

Question 15

State the end behavior of the function. f(x) = -3x⁵ + 5x⁴ + x²- 2x + 2

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Question 16

State the end behavior and y-intercept of the function. Do not graph. f(x) = -2x⁴ - 2x³ + x²- 2x + 5

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Question 17

Use the following to answer questions :
p(x) = xS1U1P14S1S1P0 - 5xS1U1P13S1S1P0 + 5xS1U1P12S1S1P0 + 5x - 6
-Apply Descartes's rule of signs to count the possible number of positive, negative, and complex roots (organize a table).

Accepted Answer

Using Descartes's rule of signs, we count the number of positive and negative roots as follows: - The number of sign changes in p(x) is 2 (from +5x to -6, and from -5x³ to +5x²), so there are at most 2 positive roots. - The number of sign changes in p(-x) is 1 (from -5x³ to +5x²), so there is at most 1 negative root. - The number of non-real roots (complex roots) is either 0 or even (because the coefficients are all real). Therefore, the best choice is C since it satisfies all of these conditions.

Question 18

Sketch the graph of the function using the degree, end behavior, x- and y-intercepts, zeroes of multiplicity, and a few midinterval points to round out the graph. Connect all points with a smooth, continuous curve. f(x) = (x - 1)³(x + 2)(x + 3)

Accepted Answer

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Question 19

Use Descartes's rule of signs to determine the possible combinations of real and complex roots for the polynomial. Then graph the function on the standard window of a graphing calculator and adjust it as needed until you're certain all real roots are in clear view. Use this screen and a list of the possible rational zeroes (rational roots theorem) to factor the polynomial and find all zeroes (real and complex). p(x) = 20x³ - 153x² - 62x + 48

Accepted Answer

Descartes's rule of signs and the rational roots theorem, along with graphing, help identify the real roots and their multiplicities, leading to the correct factorization in option C.

Question 20

For the complex polynomial below, one of its zeroes is x = 3 - i. Use the given zero to help find all zeroes of the polynomial, then write the polynomial in completely factored form. Hint: synthetic division and the quadratic formula can be applied to all polynomials, even those with complex coefficients. C(x) = x³ - 3x² + (3 + 3i)x + (-6 + 2i)

Accepted Answer

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Quiz 3: Polynomial and Rational Functions

Find all zeroes (real and complex) of the polynomial. p(x) = 4x³ - 5x² - 23x + 6

Gather information on the polynomial using the rational roots theorem, testing for 1 and -1, Descartes's rule of signs, and the upper and lower bounds property. Respond explicitly to each. p(x) = x⁴ - 3x³ + 9x - 27

State the degree and end behavior of the function. f(x) = (x - 1) ⁶(x + 3) (x - 3) ³

State the end behavior and y-intercept of the function. Do not graph. f(x) = x⁵ + 5x⁴ - 5x²- 3x + 5

Use the following to answer questions : (Gridlines are spaced one unit apart.) -Find the minimum possible degree of the function and write it in factored form. Assume all zeroes are real.

Use the following to answer questions : (Gridlines are spaced one unit apart.) -State whether the degree of the function is even or odd.

Use the following to answer questions : (Gridlines are spaced one unit apart.) -Use the graph to estimate the zeroes of the function, then state whether their multiplicity is even or odd.

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial. f(x) = x³ - 3x - 2

State the degree, end behavior, and y-intercept of the function. f(x) = -(x - 5)(x - 1)⁸(x - 2)²

State the end behavior of the function. f(x) = 5x⁴ + 2x³ - 5x²- 5x - 3

Use the following to answer questions : p(x) = x⁴ - 5x³ + 5x² + 5x - 6 -Use the information gathered, testing for 1 and -1, synthetic division, and the factor theorem to factor p completely.

Find all zeroes (real and complex) of the polynomial. p(x) = 6x³ + 29x² - 149x + 24

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial. f(x) = x³ - 4x²+ x + 6

State the end behavior of the function. f(x) = -3x⁵ + 5x⁴ + x²- 2x + 2

State the end behavior and y-intercept of the function. Do not graph. f(x) = -2x⁴ - 2x³ + x²- 2x + 5

Use the following to answer questions : p(x) = x⁴ - 5x³ + 5x² + 5x - 6 -Apply Descartes's rule of signs to count the possible number of positive, negative, and complex roots (organize a table) .

Sketch the graph of the function using the degree, end behavior, x- and y-intercepts, zeroes of multiplicity, and a few midinterval points to round out the graph. Connect all points with a smooth, continuous curve. f(x) = (x - 1)³(x + 2)(x + 3)

Quiz 3: Polynomial and Rational Functions

Find all zeroes (real and complex) of the polynomial. p(x) = 4x3 - 5x2 - 23x + 6

Gather information on the polynomial using the rational roots theorem, testing for 1 and -1, Descartes's rule of signs, and the upper and lower bounds property. Respond explicitly to each. p(x) = x4 - 3x3 + 9x - 27

State the degree and end behavior of the function. f(x) = (x - 1) 6(x + 3) (x - 3) 3

State the end behavior and y-intercept of the function. Do not graph. f(x) = x5 + 5x4 - 5x2 - 3x + 5

Use the following to answer questions : (Gridlines are spaced one unit apart.) -Find the minimum possible degree of the function and write it in factored form. Assume all zeroes are real.

Use the following to answer questions : (Gridlines are spaced one unit apart.) -State whether the degree of the function is even or odd.

Use the following to answer questions : (Gridlines are spaced one unit apart.) -Use the graph to estimate the zeroes of the function, then state whether their multiplicity is even or odd.

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial. f(x) = x3 - 3x - 2

State the degree, end behavior, and y-intercept of the function. f(x) = -(x - 5)(x - 1)8(x - 2)2

State the end behavior of the function. f(x) = 5x4 + 2x3 - 5x2 - 5x - 3

Use the following to answer questions : p(x) = x4 - 5x3 + 5x2 + 5x - 6 -Use the information gathered, testing for 1 and -1, synthetic division, and the factor theorem to factor p completely.

Find all zeroes (real and complex) of the polynomial. p(x) = 6x3 + 29x2 - 149x + 24

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial. f(x) = x3 - 4x2 + x + 6

State the end behavior of the function. f(x) = -3x5 + 5x4 + x2 - 2x + 2

State the end behavior and y-intercept of the function. Do not graph. f(x) = -2x4 - 2x3 + x2 - 2x + 5

Use the following to answer questions : p(x) = x4 - 5x3 + 5x2 + 5x - 6 -Apply Descartes's rule of signs to count the possible number of positive, negative, and complex roots (organize a table) .

Sketch the graph of the function using the degree, end behavior, x- and y-intercepts, zeroes of multiplicity, and a few midinterval points to round out the graph. Connect all points with a smooth, continuous curve. f(x) = (x - 1)3(x + 2)(x + 3)

Find all zeroes (real and complex) of the polynomial. p(x) = 4x³ - 5x² - 23x + 6

Gather information on the polynomial using the rational roots theorem, testing for 1 and -1, Descartes's rule of signs, and the upper and lower bounds property. Respond explicitly to each. p(x) = x⁴ - 3x³ + 9x - 27

State the degree and end behavior of the function. f(x) = (x - 1) ⁶(x + 3) (x - 3) ³

State the end behavior and y-intercept of the function. Do not graph. f(x) = x⁵ + 5x⁴ - 5x²- 3x + 5

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial. f(x) = x³ - 3x - 2

State the degree, end behavior, and y-intercept of the function. f(x) = -(x - 5)(x - 1)⁸(x - 2)²

State the end behavior of the function. f(x) = 5x⁴ + 2x³ - 5x²- 5x - 3

Use the following to answer questions : p(x) = x⁴ - 5x³ + 5x² + 5x - 6 -Use the information gathered, testing for 1 and -1, synthetic division, and the factor theorem to factor p completely.

Find all zeroes (real and complex) of the polynomial. p(x) = 6x³ + 29x² - 149x + 24

Use the Guidelines for Graphing Polynomial Functions to graph the polynomial. f(x) = x³ - 4x²+ x + 6

State the end behavior of the function. f(x) = -3x⁵ + 5x⁴ + x²- 2x + 2

State the end behavior and y-intercept of the function. Do not graph. f(x) = -2x⁴ - 2x³ + x²- 2x + 5

Use the following to answer questions : p(x) = x⁴ - 5x³ + 5x² + 5x - 6 -Apply Descartes's rule of signs to count the possible number of positive, negative, and complex roots (organize a table) .

Sketch the graph of the function using the degree, end behavior, x- and y-intercepts, zeroes of multiplicity, and a few midinterval points to round out the graph. Connect all points with a smooth, continuous curve. f(x) = (x - 1)³(x + 2)(x + 3)