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1.15 Assignment: Polynomial and rational functions Fadia Yaseen Question 1. Rod, might be thinking of these two different polynomial expressions that satisfy the characteristics he provided: Polynomial 1 Degree: 4 Number of terms: 3 One term with degree 1: -5x One term with a coefficient of -5: -5x Y-intercept at 10: 10 Polynomial 1: Polynomial 2 Degree: 4 Number of terms: 3 One term with degree 1: -5x One term with a coefficient of -5: -5x Y-intercept at 10: 10 Polynomial 2: These are two valid polynomial expressions that fulfil all the given conditions. However, there are other polynomials that might also satisfy these conditions by varying the coefficients and terms, so these are not the only possible answers. Question 2. Think of a one-variable polynomial expression with the following characteristics: a. Degree of 3 b. Consists of four terms c. One term has a degree of 2 d. One term has a coefficient of -3 e. Has a y-intercept at 8 To check a student's response to this question, the teacher can follow these steps: 1. Examine the Polynomial Expressions: First, review the two polynomial expressions provided by the student to ensure they meet the specified criteria. 2. Degree of 3: Verify that both polynomials have a degree of 3, which means the highest power of the variable in the polynomial should be 3. 3. Consists of Four Terms: Count the number of terms in each polynomial to ensure they consist of four terms. Each term should be separated by addition or subtraction.

4. One Term with Degree of 2: Check if at least one term in each polynomial has a degree of 2 (the variable raised to the power of 2). 5. One Term with Coefficient of -3: Confirm that at least one term in each polynomial has a coefficient of -3. This means a constant multiplied by the variable. 6. Y-Intercept at 8: Evaluate both polynomials by setting x = 0 and ensure that the result is 8. This represents the y-intercept. 7. Ensure No Additional Terms: Make sure there are no additional terms in the polynomials that violate the given conditions. 8. Allow for Variation: Keep in mind that there may be multiple valid answers. The student could have chosen different coefficients, variables, or constant terms that still satisfy the requirements. By following these steps, the teacher can verify if the student's response is correct and if it meets all the specified criteria. This approach allows the teacher to assess the student's understanding of polynomial characteristics and their ability to create valid polynomials based on the given conditions. Question 3. A function that satisfies the given domain and range criteria is: This function's domain is { 𝑥𝑥 ∈ 𝑅𝑅 | 2 ≤ 𝑥𝑥 ≤ 10} and its range is { 𝑦𝑦 ∈ 𝑅𝑅 | 5 ≤ 𝑦𝑦 ≤ 10}. It's a linear function with a positive slope, so as x varies within the specified domain, y varies within the specified range. Question 4. Example 1: Step 1: Calculate the function values for x =1,2,3,4,... Step 2: Calculate the constant finite differences: First differences: 10 - 3 = 7, 21 - 10 = 11, 36 - 21 = 15 Second differences: 11 - 7 = 4, 15 - 11 = 4 In this example, as "a" (which is 2) and "n" (which is 2) are constant, the second differences are also constant, which suggests a quadratic relationship. Example 2: Step 1: Calculate the function values for x =1,2,3,4,...

Step 2: Calculate the constant finite differences: First differences: 26 - 4 = 22, 84 - 26 = 58, 184 - 84 = 100 Second differences: 58 - 22 = 36, 100 - 58 = 42 In this example, as "a" (which is 3) and "n" (which is 3) are constant, the second differences are also constant, which suggests a cubic relationship. Example 3: Step 1: Calculate the function values for x =1,2,3,4,... Step 2: Calculate the constant finite differences: First differences: 82 - 6 = 76, 456 - 82 = 374, 1700 - 456 = 1244 Second differences: 374 - 76 = 298, 1244 - 374 = 870 In this example, as "a" (which is 5) and "n" (which is 4) are constant, the second differences are also constant, which suggests a quartic (fourth-degree) relationship. In all three examples, we observe that when "a" and "n" remain constant, the second differences between consecutive function values are also constant. This pattern suggests that the degree of the polynomial and the value of "a" both have an impact on the constant finite differences. This observation reinforces the idea that the degree of the polynomial corresponds to the order of differences that remain constant. Specifically, for a polynomial of degree "n," the "n"-th differences will remain constant when the coefficients are held constant. Question 5. a) I used the Rational Root Theorem to find potential rational roots. I found x=1 as a root and x=10 as a root. After synthetic division by (x-1) and (x-10), I was left with the quadratic factor 2x-3. So, the factorization of the polynomial is

b) I first found that x=1 as a root based on the Rational Root Theorem. After synthetic division by (x-1), we were left with the quadratic factor . The quadratic factor does not have real roots since its discriminant is negative. Therefore, the factorization of the polynomial is Question 6. Step 1: Find the x-intercepts by setting Step 2: Determine the y-intercept at (0, 20). Step 3: Analyse the behaviour at these key points: Step 4: Consider the end behaviour: Step 5: Sketch the graph with two humps, one at x=-1 and another at x=4, a root at (x=5), and end behaviour as described. Question 7.

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