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Polynomial Analysis

Grade 12 · Algebra · Worksheet 1

  1. lim(x→∞) (4x⁵ - 3x⁴ + 7x² - 9)/(2x⁵ - 5x³ + 8) = ? Answer: ______________
  2. f(x) = -8x⁹ + 12x⁵ - 7x³ + 4. Describe the end behavior of f(x) as x → ∞ and as x → -∞. Answer: ______________
  3. Aroha is studying the long-term behavior of a polynomial function that models the profit (in thousands of dollars) of a company over time, where x represents the number of years since the company was founded. The profit function is P(x) = -7x^9 + 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. Noah graphs the polynomial function f(x) = 6x^7 - 11x^4 + 16x - 21 on a coordinate plane. Describe the end behavior of this polynomial as x approaches negative infinity and as x approaches positive infinity. Answer: ______________
  5. f(x) = -6x¹¹ + 21x⁶ - 4x³ + 9. Describe the end behavior as x → ∞ and x → -∞. Answer: ______________
  6. f(x) = -2x⁶ + 8x⁴ - 10x² + 6. Describe the end behavior as x → ∞ and x → -∞. Answer: ______________
  7. Matiu is an environmental scientist modeling the population of a rare bird species in a protected forest. The population (in hundreds) over time t (in years) is modeled by the polynomial function P(t) = -2t⁶ + 4t⁴ - 6t² + 10. As time goes on (t → ∞) and as we look back in time (t → -∞), what happens to the bird population? Describe the end behavior of P(t) using the degree and leading coefficient. Answer: ______________
  8. Sophia is analyzing the long-term behavior of a polynomial function that models the profit (in thousands of dollars) of a company over time, where x represents years since 2000. The profit function is given by P(x) = -7x^9 + 4x^5 - 2x^3 + 11. Describe the end behavior of this polynomial as x approaches positive infinity and as x approaches negative infinity. Answer: ______________
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Answer Key & Explanations

Polynomial Analysis · Grade 12 · Worksheet 1

  1. lim(x→∞) (4x⁵ - 3x⁴ + 7x² - 9)/(2x⁵ - 5x³ + 8) = ? Answer: 2 Solution: Identify the highest degree terms in numerator and denominator Numerator: 4x⁵ Denominator: 2x⁵ For large values of x, the lower degree terms become negligible compared to the highest degree terms The limit equals the ratio of the leading coefficients 4x⁵/2x⁵ = 4/2 = 2 Therefore, lim(x→∞) (4x⁵ -…
    Full step-by-step solution

    Step 1: Identify the highest degree terms in numerator and denominator Numerator: 4x⁵ Denominator: 2x⁵ Step 2: For large values of x, the lower degree terms become negligible compared to the highest degree terms Step 3: The limit equals the ratio of the leading coefficients 4x⁵/2x⁵ = 4/2 = 2 Step 4: Therefore, lim(x→∞) (4x⁵ - 3x⁴ + 7x² - 9)/(2x⁵ - 5x³ + 8) = 2 The answer is 2.

  2. f(x) = -8x⁹ + 12x⁵ - 7x³ + 4. Describe the end behavior of f(x) as x → ∞ and as x → -∞. Answer: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞ Solution: Identify the leading term. The leading term is -8x⁹ because it has the highest exponent (9). Determine the degree.
    Full step-by-step solution

    Step 1: Identify the leading term. The leading term is -8x⁹ because it has the highest exponent (9). Step 2: Determine the degree. The degree is 9, which is odd. Step 3: Determine the leading coefficient. The leading coefficient is -8, which is negative. Step 4: For an odd-degree polynomial with a negative leading coefficient: - As x → ∞ (right end), f(x) → -∞ (the graph falls to the right) - As x → -∞ (left end), f(x) → ∞ (the graph rises to the left) The answer is: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞.

  3. Aroha is studying the long-term behavior of a polynomial function that models the profit (in thousands of dollars) of a company over time, where x represents the number of years since the company was founded. The profit function is P(x) = -7x^9 + 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 approaches positive infinity, P(x) approaches negative infinity. As x approaches negative infinity, P(x) approaches positive infinity. Solution: Identify the leading term. The polynomial is P(x) = -7x^9 + 4x^5 - 2x^3 + 11. The leading term is the term with the highest exponent, which is -7x^9.
    Full step-by-step solution

    Step 1: Identify the leading term. The polynomial is P(x) = -7x^9 + 4x^5 - 2x^3 + 11. The leading term is the term with the highest exponent, which is -7x^9. The degree is 9 (odd), and the leading coefficient is -7 (negative). Step 2: Analyze end behavior as x approaches positive infinity. For very large positive x, x^9 is a very large positive number. Multiplying by -7 makes it a very large negative number. Therefore, as x → +∞, P(x) → -∞. Step 3: Analyze end behavior as x approaches negative infinity. For very large negative x, x^9 is a very large negative number (since an odd power preserves the sign). Multiplying by -7 makes it a very large positive number. Therefore, as x → -∞, P(x) → +∞. Final Answer: As x → +∞, P(x) → -∞. As x → -∞, P(x) → +∞.

  4. Noah graphs the polynomial function f(x) = 6x^7 - 11x^4 + 16x - 21 on a coordinate plane. Describe the end behavior of this polynomial as x approaches negative infinity and as x approaches positive infinity. Answer: As x → -∞, f(x) → -∞; as x → +∞, f(x) → +∞ Solution: Identify the leading term of f(x) = 6x^7 - 11x^4 + 16x - 21. The leading term is the term with the highest exponent, which is 6x^7. Determine the degree.
    Full step-by-step solution

    Step 1: Identify the leading term of f(x) = 6x^7 - 11x^4 + 16x - 21. The leading term is the term with the highest exponent, which is 6x^7. Step 2: Determine the degree. The exponent of the leading term is 7, which is odd. Step 3: Determine the leading coefficient. The coefficient of the leading term is 6, which is positive. Step 4: Recall the end behavior rules for polynomials: - For odd degree with positive leading coefficient: as x → -∞, f(x) → -∞; as x → +∞, f(x) → +∞. - For odd degree with negative leading coefficient: as x → -∞, f(x) → +∞; as x → +∞, f(x) → -∞. Step 5: Apply the rule: Since the degree is odd (7) and the leading coefficient is positive (6), the end behavior is: as x → -∞, f(x) → -∞; as x → +∞, f(x) → +∞. Therefore, the end behavior is: as x approaches negative infinity, f(x) approaches negative infinity; as x approaches positive infinity, f(x) approaches positive infinity.

  5. f(x) = -6x¹¹ + 21x⁶ - 4x³ + 9. Describe the end behavior as x → ∞ and x → -∞. Answer: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞ Solution: Identify the leading term. The leading term is -6x¹¹ because it has the highest exponent (11). Determine the degree.
    Full step-by-step solution

    Step 1: Identify the leading term. The leading term is -6x¹¹ because it has the highest exponent (11). Step 2: Determine the degree. The degree is 11, which is odd. Step 3: Determine the leading coefficient. The leading coefficient is -6, which is negative. Step 4: For an odd-degree polynomial with a negative leading coefficient: - As x → ∞, the leading term -6x¹¹ → -∞ (since a large positive number raised to an odd power is positive, multiplied by -6 gives negative infinity). So f(x) → -∞. - As x → -∞, the leading term -6x¹¹ → ∞ (since a large negative number raised to an odd power is negative, multiplied by -6 gives positive infinity). So f(x) → ∞. The answer is: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞.

  6. f(x) = -2x⁶ + 8x⁴ - 10x² + 6. Describe the end behavior as x → ∞ and x → -∞. Answer: As x → ∞, f(x) → -∞; as x → -∞, f(x) → -∞ Solution: Identify the leading term. The polynomial is f(x) = -2x⁶ + 8x⁴ - 10x² + 6. The highest power is x⁶, so the leading term is -2x⁶.
    Full step-by-step solution

    Step 1: Identify the leading term. The polynomial is f(x) = -2x⁶ + 8x⁴ - 10x² + 6. The highest power is x⁶, so the leading term is -2x⁶. Step 2: Determine the degree. The degree is 6, which is even. Step 3: Determine the leading coefficient. The leading coefficient is -2, which is negative. Step 4: Apply end behavior rules for polynomials: - Even degree: both ends go in the same direction. - Negative leading coefficient: both ends go downward (toward -∞). Step 5: Therefore: As x → ∞, f(x) → -∞ As x → -∞, f(x) → -∞ The answer is: As x → ∞, f(x) → -∞; as x → -∞, f(x) → -∞

  7. Matiu is an environmental scientist modeling the population of a rare bird species in a protected forest. The population (in hundreds) over time t (in years) is modeled by the polynomial function P(t) = -2t⁶ + 4t⁴ - 6t² + 10. As time goes on (t → ∞) and as we look back in time (t → -∞), what happens to the bird population? Describe the end behavior of P(t) using the degree and leading coefficient. Answer: As t → ∞, P(t) → -∞; as t → -∞, P(t) → -∞ Solution: Identify the leading term of P(t) = -2t⁶ + 4t⁴ - 6t² + 10. The leading term is -2t⁶, because it has the highest exponent. Determine the degree: the exponent of the leading term is 6, which is an even number.
    Full step-by-step solution

    Step 1: Identify the leading term of P(t) = -2t⁶ + 4t⁴ - 6t² + 10. The leading term is -2t⁶, because it has the highest exponent. Step 2: Determine the degree: the exponent of the leading term is 6, which is an even number. Step 3: Determine the leading coefficient: it is -2, which is negative. Step 4: For a polynomial with an even degree and a negative leading coefficient, both ends of the graph go downward. As t → ∞ (moving forward in time), P(t) → -∞ (population declines without bound). As t → -∞ (looking backward in time), P(t) → -∞ (population also declines without bound). Step 5: Final answer: As t → ∞, P(t) → -∞; as t → -∞, P(t) → -∞.

  8. Sophia is analyzing the long-term behavior of a polynomial function that models the profit (in thousands of dollars) of a company over time, where x represents years since 2000. The profit function is given by P(x) = -7x^9 + 4x^5 - 2x^3 + 11. Describe the end behavior of this polynomial as x approaches positive infinity and as x approaches negative infinity. Answer: As x → +∞, P(x) → -∞; as x → -∞, P(x) → +∞ Solution: Identify the leading term of the polynomial P(x) = -7x^9 + 4x^5 - 2x^3 + 11. The leading term is the term with the highest power of x, which is -7x^9. Determine the degree of the polynomial.
    Full step-by-step solution

    Step 1: Identify the leading term of the polynomial P(x) = -7x^9 + 4x^5 - 2x^3 + 11. The leading term is the term with the highest power of x, which is -7x^9. Step 2: Determine the degree of the polynomial. The degree is 9, which is odd. Step 3: Determine the leading coefficient. The leading coefficient is -7, which is negative. Step 4: For a polynomial with an odd degree and a negative leading coefficient, the end behavior is: as x → +∞, P(x) → -∞; as x → -∞, P(x) → +∞. Reasoning: For large positive x, -7x^9 is a large negative number (since a positive number to an odd power is positive, multiplied by -7 gives negative). For large negative x, x^9 is negative (since a negative number to an odd power is negative), and -7 times a negative gives a large positive number. Thus, the end behavior is: as x → +∞, P(x) → -∞; as x → -∞, P(x) → +∞.