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Polynomial Analysis

Grade 12 Ā· Algebra Ā· Worksheet 2

  1. lim(xā†’āˆž) (4xāµ - 3xĀ³ + 7x - 2)/(2xāµ + 5xĀ² - 9) = ? Answer: ______________
  2. Noah is analyzing the long-term behavior of a polynomial function that models the profit (in thousands of dollars) of a tech startup over time, where x represents years since the company's founding. The function is f(x) = 7xā¶ - 9xĀ³ + 2x - 5. Describe the end behavior of this polynomial as x ā†’ āˆž and as x ā†’ -āˆž. Answer: ______________
  3. Charlotte is analyzing the long-term behavior of a polynomial function that models the growth of a certain biological population over time. The function is f(x) = -9x^7 + 4x^5 - 2x^3 + 11. Describe the end behavior of this polynomial function as x approaches positive infinity and as x approaches negative infinity. Answer: ______________
  4. f(x) = -7xā¹ + 13xāµ - 3xĀ³ + 11. Describe the end behavior of f(x) as x ā†’ āˆž and as x ā†’ -āˆž. Answer: ______________
  5. lim(xā†’āˆž) (4xāµ - 3xā“ + 7xĀ² - 9)/(5xāµ - 2xĀ³ + 6) = ? Answer: ______________
  6. lim(xā†’āˆž) (3xĀ³ - 2xĀ² + 5x - 7)/(4xĀ³ + x - 1) = ? Answer: ______________
  7. A polynomial function f(x) = 3x^5 - 2x^4 + 7x^2 - 8 is graphed on a coordinate plane. The graph shows the function approaching negative infinity as x approaches negative infinity and approaching positive infinity as x approaches positive infinity. What is the degree of this polynomial and what is the sign of its leading coefficient? Answer: ______________
  8. Emma is studying the flight path of a model rocket. The height of the rocket, in meters, after t seconds is modeled by the polynomial function h(t) = -5tāµ + 25tĀ³ - 10t. Emma wants to describe the long-term behavior of the rocket's height as time increases without bound (t ā†’ āˆž) and as time goes backward (t ā†’ -āˆž). Determine the end behavior of h(t) using the leading term. Answer: ______________
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Answer Key & Explanations

Polynomial Analysis Ā· Grade 12 Ā· Worksheet 2

  1. lim(xā†’āˆž) (4xāµ - 3xĀ³ + 7x - 2)/(2xāµ + 5xĀ² - 9) = ? Answer: 2 Solution: Numerator degree: 5 (from 4xāµ) Denominator degree: 5 (from 2xāµ) Since degrees are equal, the limit is the ratio of leading coefficients Leading coefficient of numerator: 4 Leading coefficient of denominator: 2 4 Ć· 2 = 2 Therefore, lim(xā†’āˆž) (4xāµ - 3xĀ³ + 7x - 2)/(2xāµ + 5xĀ² - 9) = 2
    Full step-by-step solution

    Step 1: Identify the degrees of numerator and denominator Numerator degree: 5 (from 4xāµ) Denominator degree: 5 (from 2xāµ) Step 2: Since degrees are equal, the limit is the ratio of leading coefficients Leading coefficient of numerator: 4 Leading coefficient of denominator: 2 Step 3: Calculate the ratio 4 Ć· 2 = 2 Step 4: Therefore, lim(xā†’āˆž) (4xāµ - 3xĀ³ + 7x - 2)/(2xāµ + 5xĀ² - 9) = 2

  2. Noah is analyzing the long-term behavior of a polynomial function that models the profit (in thousands of dollars) of a tech startup over time, where x represents years since the company's founding. The function is f(x) = 7xā¶ - 9xĀ³ + 2x - 5. Describe the end behavior of this polynomial as x ā†’ āˆž and as x ā†’ -āˆž. Answer: As x ā†’ āˆž, f(x) ā†’ āˆž; as x ā†’ -āˆž, f(x) ā†’ āˆž Solution: Identify the leading term. The polynomial is f(x) = 7xā¶ - 9xĀ³ + 2x - 5. The term with the highest exponent is 7xā¶, so the leading term is 7xā¶.
    Full step-by-step solution

    Step 1: Identify the leading term. The polynomial is f(x) = 7xā¶ - 9xĀ³ + 2x - 5. The term with the highest exponent is 7xā¶, so the leading term is 7xā¶. Step 2: Determine the degree. The exponent of the leading term is 6, which is an even number. Step 3: Determine the leading coefficient. The coefficient of the leading term is 7, which is positive. Step 4: Recall the end behavior rules for polynomials: - If the degree is even and the leading coefficient is positive, as x ā†’ āˆž, f(x) ā†’ āˆž, and as x ā†’ -āˆž, f(x) ā†’ āˆž. - If the degree is even and the leading coefficient is negative, as x ā†’ āˆž, f(x) ā†’ -āˆž, and as x ā†’ -āˆž, f(x) ā†’ -āˆž. - If the degree is odd and the leading coefficient is positive, as x ā†’ āˆž, f(x) ā†’ āˆž, and as x ā†’ -āˆž, f(x) ā†’ -āˆž. - If the degree is odd and the leading coefficient is negative, as x ā†’ āˆž, f(x) ā†’ -āˆž, and as x ā†’ -āˆž, f(x) ā†’ āˆž. Step 5: Apply the rule. Since the degree (6) is even and the leading coefficient (7) is positive, both ends of the graph go upward. Therefore, as x ā†’ āˆž, f(x) ā†’ āˆž, and as x ā†’ -āˆž, f(x) ā†’ āˆž. Final answer: As x ā†’ āˆž, f(x) ā†’ āˆž; as x ā†’ -āˆž, f(x) ā†’ āˆž.

  3. Charlotte is analyzing the long-term behavior of a polynomial function that models the growth of a certain biological population over time. The function is f(x) = -9x^7 + 4x^5 - 2x^3 + 11. Describe the end behavior of this polynomial function as x approaches positive infinity and as x approaches negative infinity. Answer: As x ā†’ +āˆž, f(x) ā†’ -āˆž; as x ā†’ -āˆž, f(x) ā†’ +āˆž Solution: Identify the leading term. The highest power of x is x^7, with coefficient -9. The degree is 7, which is odd.
    Full step-by-step solution

    Step 1: Identify the leading term. The highest power of x is x^7, with coefficient -9. So the leading term is -9x^7. Step 2: The degree is 7, which is odd. For odd-degree polynomials, the ends go in opposite directions. Step 3: The leading coefficient is -9, which is negative. For an odd degree with a negative leading coefficient, as x ā†’ +āˆž, f(x) ā†’ -āˆž; as x ā†’ -āˆž, f(x) ā†’ +āˆž. Step 4: Therefore, the end behavior is: as x approaches positive infinity, f(x) approaches negative infinity; as x approaches negative infinity, f(x) approaches positive infinity.

  4. f(x) = -7xā¹ + 13xāµ - 3xĀ³ + 11. Describe the end behavior of f(x) as x ā†’ āˆž and as x ā†’ -āˆž. Answer: As x ā†’ āˆž, f(x) ā†’ -āˆž; as x ā†’ -āˆž, f(x) ā†’ āˆž Solution: Identify the leading term. The leading term is -7xā¹ because it has the highest exponent. Determine the degree.
    Full step-by-step solution

    Step 1: Identify the leading term. The leading term is -7xā¹ because it has the highest exponent. Step 2: Determine the degree. The degree is 9, which is odd. Step 3: Determine the leading coefficient. The leading coefficient is -7, which is negative. Step 4: Apply end behavior rules for odd degree with negative leading coefficient: - As x ā†’ āˆž (right end), f(x) ā†’ -āˆž (downward) - As x ā†’ -āˆž (left end), f(x) ā†’ āˆž (upward) Step 5: Write the final description. The answer is: As x ā†’ āˆž, f(x) ā†’ -āˆž; as x ā†’ -āˆž, f(x) ā†’ āˆž

  5. lim(xā†’āˆž) (4xāµ - 3xā“ + 7xĀ² - 9)/(5xāµ - 2xĀ³ + 6) = ? Answer: 4/5 Solution: Identify the degrees of the polynomials. Both numerator and denominator are degree 5. Compare the leading coefficients.
    Full step-by-step solution

    Step 1: Identify the degrees of the polynomials. Both numerator and denominator are degree 5. Step 2: Compare the leading coefficients. The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 5. Step 3: Since the degrees are equal, the limit as x approaches infinity is the ratio of the leading coefficients. Step 4: Calculate the ratio: 4/5. The answer is 4/5.

  6. lim(xā†’āˆž) (3xĀ³ - 2xĀ² + 5x - 7)/(4xĀ³ + x - 1) = ? Answer: 3/4 Solution: lim(xā†’āˆž) (3xĀ³ - 2xĀ² + 5x - 7) / (4xĀ³ + x - 1) Identify the highest power of x in the denominator Both numerator and denominator are polynomials. The highest power of x in both is xĀ³.
    Full step-by-step solution

    Let's find the limit step by step. We want: lim(xā†’āˆž) (3xĀ³ - 2xĀ² + 5x - 7) / (4xĀ³ + x - 1) --- **Step 1: Identify the highest power of x in the denominator** Both numerator and denominator are polynomials. The highest power of x in both is xĀ³. --- **Step 2: Divide numerator and denominator by xĀ³** Numerator: (3xĀ³ - 2xĀ² + 5x - 7) / xĀ³ = 3 - 2/x + 5/xĀ² - 7/xĀ³ Denominator: (4xĀ³ + x - 1) / xĀ³ = 4 + 1/xĀ² - 1/xĀ³ So the expression becomes: [3 - 2/x + 5/xĀ² - 7/xĀ³] / [4 + 1/xĀ² - 1/xĀ³] --- **Step 3: Take the limit as x ā†’ āˆž** As x ā†’ āˆž: - 2/x ā†’ 0 - 5/xĀ² ā†’ 0 - 7/xĀ³ ā†’ 0 - 1/xĀ² ā†’ 0 - 1/xĀ³ ā†’ 0 So the expression approaches: (3 - 0 + 0 - 0) / (4 + 0 - 0) = 3/4 --- **Step 4: Conclusion** lim(xā†’āˆž) (3xĀ³ - 2xĀ² + 5x - 7)/(4xĀ³ + x - 1) = 3/4 --- **Final answer:** 3/4

  7. A polynomial function f(x) = 3x^5 - 2x^4 + 7x^2 - 8 is graphed on a coordinate plane. The graph shows the function approaching negative infinity as x approaches negative infinity and approaching positive infinity as x approaches positive infinity. What is the degree of this polynomial and what is the sign of its leading coefficient? Answer: degree: 5, leading coefficient: positive Solution: Analyze the given end behavior: As x ā†’ -āˆž, f(x) ā†’ -āˆž and As x ā†’ āˆž, f(x) ā†’ āˆž - If leading coefficient is positive: left end goes down, right end goes up - If leading coefficient is negative: left end goes up, right end goes down The given behavior (down on left, up on right) matches an odd-degreeā€¦
    Full step-by-step solution

    Step 1: Analyze the given end behavior: As x ā†’ -āˆž, f(x) ā†’ -āˆž and As x ā†’ āˆž, f(x) ā†’ āˆž Step 2: Recall that for polynomials with odd degree: - If leading coefficient is positive: left end goes down, right end goes up - If leading coefficient is negative: left end goes up, right end goes down Step 3: The given behavior (down on left, up on right) matches an odd-degree polynomial with positive leading coefficient Step 4: The polynomial f(x) = 3x^5 - 2x^4 + 7x^2 - 8 has degree 5 (odd) and leading coefficient 3 (positive) Step 5: This confirms our analysis from the end behavior Therefore, the degree is 5 and the leading coefficient is positive.

  8. Emma is studying the flight path of a model rocket. The height of the rocket, in meters, after t seconds is modeled by the polynomial function h(t) = -5tāµ + 25tĀ³ - 10t. Emma wants to describe the long-term behavior of the rocket's height as time increases without bound (t ā†’ āˆž) and as time goes backward (t ā†’ -āˆž). Determine the end behavior of h(t) using the leading term. Answer: As x ā†’ āˆž, h(t) ā†’ -āˆž; as x ā†’ -āˆž, h(t) ā†’ āˆž Solution: Identify the leading term. The polynomial is h(t) = -5tāµ + 25tĀ³ - 10t. The term with the highest exponent is -5tāµ.
    Full step-by-step solution

    Step 1: Identify the leading term. The polynomial is h(t) = -5tāµ + 25tĀ³ - 10t. The term with the highest exponent is -5tāµ. So the leading term is -5tāµ. Step 2: Determine the degree. The exponent of the leading term is 5, which is odd. Step 3: Determine the leading coefficient. The coefficient of the leading term is -5, which is negative. Step 4: Apply end behavior rules. For a polynomial with odd degree and negative leading coefficient: - As t ā†’ āˆž (right side), the function goes to -āˆž. - As t ā†’ -āˆž (left side), the function goes to āˆž. Thus, the end behavior is: as t ā†’ āˆž, h(t) ā†’ -āˆž; as t ā†’ -āˆž, h(t) ā†’ āˆž.