Polynomial Analysis

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Polynomial Analysis · Grade 12 · Worksheet 3

1. lim(x→∞) (3x⁴ - 2x³ + 5x - 1)/(2x⁴ + x² - 7) = ? Answer: 3/2 Solution: lim(x→∞) (3x⁴ - 2x³ + 5x - 1) / (2x⁴ + x² - 7) Identify the highest power of x in the denominator The denominator is 2x⁴ + x² - 7. The highest power of x here is x⁴.
   Full step-by-step solution

   Let's find the limit step by step. We want: lim(x→∞) (3x⁴ - 2x³ + 5x - 1) / (2x⁴ + x² - 7) --- **Step 1: Identify the highest power of x in the denominator** The denominator is 2x⁴ + x² - 7. The highest power of x here is x⁴. --- **Step 2: Divide numerator and denominator by x⁴** This is a standard technique for limits at infinity of rational functions. Numerator divided by x⁴: (3x⁴ - 2x³ + 5x - 1) / x⁴ = 3 - 2/x + 5/x³ - 1/x⁴ Denominator divided by x⁴: (2x⁴ + x² - 7) / x⁴ = 2 + 1/x² - 7/x⁴ So the expression becomes: [3 - 2/x + 5/x³ - 1/x⁴] / [2 + 1/x² - 7/x⁴] --- **Step 3: Take the limit as x → ∞** As x → ∞, terms with x in the denominator go to 0: - 2/x → 0 - 5/x³ → 0 - 1/x⁴ → 0 - 1/x² → 0 - 7/x⁴ → 0 So the limit becomes: (3 - 0 + 0 - 0) / (2 + 0 - 0) = 3/2 --- **Step 4: Conclusion** The limit is 3/2. --- ANSWER: 3/2

2. Sophia is studying the flight path of a model rocket. The height of the rocket above the ground, in meters, after t seconds is modeled by the polynomial function h(t) = -7t^4 + 9t^3 + 12t^2 + 5t. Describe the end behavior of this polynomial function as t approaches positive and negative infinity. Answer: As t → -∞, h(t) → -∞; as t → +∞, h(t) → -∞ Solution: Identify the leading term of h(t) = -7t^4 + 9t^3 + 12t^2 + 5t. The leading term is -7t^4. The degree is 4, which is even.
   Full step-by-step solution

   Step 1: Identify the leading term of h(t) = -7t^4 + 9t^3 + 12t^2 + 5t. The leading term is -7t^4. Step 2: The degree is 4, which is even. For an even-degree polynomial, both ends of the graph go in the same direction. Step 3: The leading coefficient is -7, which is negative. For a negative leading coefficient, as x → ±∞, the function goes to -∞. Step 4: Therefore, as t → -∞, h(t) → -∞ and as t → +∞, h(t) → -∞. The final answer is: As t → -∞, h(t) → -∞; as t → +∞, h(t) → -∞.

3. A polynomial function f(x) = 3x^5 - 2x^4 + 7x^2 - 5 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. The function has exactly four real roots at x = -2, x = -1, x = 1, and x = 2, and the graph crosses the x-axis at all four roots. What is the degree of this polynomial? Answer: 5 Solution: Analyze the end behavior described: as x → -∞, f(x) → -∞ and as x → +∞, f(x) → +∞ This end behavior pattern (down on left, up on right) indicates an odd-degree polynomial with a positive leading coefficient Count the number of real roots given: x = -2, x = -1, x = 1, x = 2 (four distinct real…
   Full step-by-step solution

   Step 1: Analyze the end behavior described: as x → -∞, f(x) → -∞ and as x → +∞, f(x) → +∞ Step 2: This end behavior pattern (down on left, up on right) indicates an odd-degree polynomial with a positive leading coefficient Step 3: Count the number of real roots given: x = -2, x = -1, x = 1, x = 2 (four distinct real roots) Step 4: Since the graph crosses the x-axis at all four roots, each root has odd multiplicity (at least 1) Step 5: The minimum degree needed to have four real roots is 4, but the end behavior requires an odd degree Step 6: Therefore, the degree must be at least 5 to satisfy both conditions (odd degree and at least four real roots) Step 7: The degree of the polynomial is 5 The answer is 5.

4. Kaia sketches the graph of the polynomial function f(x) = 12x^9 - 20x^7 + 15x^3 - 50 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. The leading term is the term with the highest power of x, which is 12x^9. Determine the degree.
   Full step-by-step solution

   Step 1: Identify the leading term. The leading term is the term with the highest power of x, which is 12x^9. Step 2: Determine the degree. The exponent of the leading term is 9, which is odd. Step 3: Determine the leading coefficient. The coefficient of the leading term is 12, which is positive. Step 4: Recall the end behavior rules for polynomials: - If the degree is odd, the left and right ends go in opposite directions. - If the leading coefficient is positive, as x → -∞, f(x) → -∞, and as x → +∞, f(x) → +∞. - If the leading coefficient is negative, as x → -∞, f(x) → +∞, and as x → +∞, f(x) → -∞. Step 5: Apply these rules: odd degree (9) and positive leading coefficient (12) mean as x → -∞, f(x) → -∞, and as x → +∞, f(x) → +∞. Thus, 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) = 7x⁷ - 2x⁵ + 12x³ - 17. 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) = 7x⁷ - 2x⁵ + 12x³ - 17. The highest power is x⁷, so the leading term is 7x⁷.
   Full step-by-step solution

   Step 1: Identify the leading term. The polynomial is f(x) = 7x⁷ - 2x⁵ + 12x³ - 17. The highest power is x⁷, so the leading term is 7x⁷. Step 2: Determine the degree. The degree is 7, which is odd. Step 3: Determine the leading coefficient. The leading coefficient is 7, which is positive. Step 4: For an odd-degree polynomial with a positive leading coefficient: as x → ∞, f(x) → ∞; as x → -∞, f(x) → -∞. The answer is: As x → ∞, f(x) → ∞; as x → -∞, f(x) → -∞.

6. Isabella is studying the long-term behavior of a polynomial function that models the trajectory of a rocket. The function is f(x) = -9x^7 + 4x^3 - 2x + 11. Describe the end behavior of this function as x approaches positive infinity and as x approaches negative infinity. Answer: As x → +∞, f(x) → -∞; as x → -∞, f(x) → +∞ Solution: Identify the leading term. The polynomial is f(x) = -9x^7 + 4x^3 - 2x + 11. The leading term is -9x^7 because it has the highest exponent.
   Full step-by-step solution

   Step 1: Identify the leading term. The polynomial is f(x) = -9x^7 + 4x^3 - 2x + 11. The leading term is -9x^7 because it has the highest exponent. Step 2: Determine the degree. The exponent of the leading term is 7, which is an odd number. Step 3: Determine the sign of the leading coefficient. The leading coefficient is -9, which is negative. Step 4: Recall the end behavior rule for odd-degree polynomials: if the leading coefficient is positive, the graph falls to the left and rises to the right; if negative, the graph rises to the left and falls to the right. Step 5: Apply to this function. Since the degree is odd (7) and the leading coefficient is negative (-9): - As x → +∞, the term -9x^7 dominates and becomes very large negative, so f(x) → -∞. - As x → -∞, x^7 is negative, and -9 times a negative is positive, so f(x) → +∞. The answer is: As x → +∞, f(x) → -∞; as x → -∞, f(x) → +∞.

7. Matiu is designing a new roller coaster for an amusement park. The vertical height (in meters) of the track above the ground as a function of horizontal distance x (in meters) from the start is modeled by the polynomial function H(x) = -7x⁶ + 4x⁴ - 2x² + 10. Matiu wants to know how the track behaves far to the left (as x → -∞) and far to the right (as x → +∞) to ensure safety and aesthetic continuity. Describe the end behavior of H(x) as x → -∞ and as x → +∞. Answer: As x → -∞, H(x) → -∞; as x → +∞, H(x) → -∞ Solution: Identify the leading term of H(x) = -7x⁶ + 4x⁴ - 2x² + 10. The highest power of x is 6, and the coefficient of x⁶ is -7. The degree is 6, which is even.
   Full step-by-step solution

   Step 1: Identify the leading term of H(x) = -7x⁶ + 4x⁴ - 2x² + 10. The highest power of x is 6, and the coefficient of x⁶ is -7. So the leading term is -7x⁶. Step 2: The degree is 6, which is even. For a polynomial with an even degree, as x → ±∞, the ends go in the same direction. Step 3: The leading coefficient is -7, which is negative. For an even degree with a negative leading coefficient, as x → +∞, the function goes to -∞, and as x → -∞, the function also goes to -∞. Step 4: Verify by considering large values: For x = 1000, -7(1000)⁶ is a huge negative number, so H(1000) → -∞. For x = -1000, (-1000)⁶ = 1000⁶, so -7(1000)⁶ is also huge negative, so H(-1000) → -∞. Final answer: As x → -∞, H(x) → -∞; as x → +∞, H(x) → -∞.