Make 3 problem questions withSolutions of Rational zeros of polynomial function.

1. Problem Question: Determine the rational zeros of the polynomial function f(x) = 2x^3 - 5x^2 + 3x + 1.

Solution: Step 1: List all possible rational zeros using the Rational Zero Theorem. The possible rational zeros are the factors of the constant term (1) divided by the factors of the leading coefficient (2). In this case, the possible rational zeros are ±1/2 and ±1.

Step 2: Use synthetic division or long division to test each possible rational zero. Start with the first possible rational zero, let's say x = 1/2.

   1/2 | 2   -5   3   1
       |     1/2  -2  1/2
       ---------------
       2   -9/2  1/2 3/2

Step 3: Check the remainder. If the remainder is zero, then x = 1/2 is a rational zero. If the remainder is not zero, move on to the next possible rational zero.

Step 4: Repeat steps 2 and 3 for the remaining possible rational zeros. Let's try x = -1.

   -1 | 2   -5   3   1
       |    -2    7  -10
       ---------------
       2   -7    10  -9

Step 5: Check the remainder. If the remainder is zero, then x = -1 is a rational zero. If the remainder is not zero, move on to the next possible rational zero.

Step 6: The rational zeros of the polynomial function f(x) = 2x^3 - 5x^2 + 3x + 1 are x = 1/2 and x = -1.

2. Problem Question: Find the rational zeros of the polynomial function g(x) = 3x^4 - 2x^3 + 5x^2 - 4x + 2.

Solution: Step 1: List all possible rational zeros using the Rational Zero Theorem. The possible rational zeros are the factors of the constant term (2) divided by the factors of the leading coefficient (3). In this case, the possible rational zeros are ±1/3, ±2/3, ±1, and ±2.

Step 2: Use synthetic division or long division to test each possible rational zero. Start with the first possible rational zero, let's say x = 1/3.

   1/3 | 3   -2   5   -4   2
       |     1/3  -1/3 2/3
       -----------------
       3   -5/3  4/3  -10/3  8/3

Step 3: Check the remainder. If the remainder is zero, then x = 1/3 is a rational zero. If the remainder is not zero, move on to the next possible rational zero.

Step 4: Repeat steps 2 and 3 for the remaining possible rational zeros. Let's try x = -1.

   -1 | 3   -2   5   -4   2
       |    -3    5  -10   14
       -----------------
       3   -5    10  -14   16

Step 5: Check the remainder. If the remainder is zero, then x = -1 is a rational zero. If the remainder is not zero, move on to the next possible rational zero.

Step 6: The rational zeros of the polynomial function g(x) = 3x^4 - 2x^3 + 5x^2 - 4x + 2 are x = 1/3 and x = -1.

3. Problem Question: Determine the rational zeros of the polynomial function h(x) = x^3 - 4x^2 + 5x - 2.

Solution: Step 1: List all possible rational zeros using the Rational Zero Theorem. The possible rational zeros are the factors of the constant term (-2) divided by the factors of the leading coefficient (1). In this case, the possible rational zeros are ±1 and ±2.

Step 2: Use synthetic division or long division to test each possible rational zero. Start with the first possible rational zero, let's say x = 1.

   1 | 1   -4   5   -2
      |     1   -3   2
      -------------
      1   -3   2    0

Step 3: Check the remainder. If the remainder is zero, then x = 1 is a rational zero. If the remainder is not zero, move on to the next possible rational zero.

Step 4: Repeat steps 2 and 3 for the remaining possible rational zeros. Let's try x = -1.

   -1 | 1   -4   5   -2
       |    -1    5  -10
       -------------
       1   -5   10   -12

Step 5: Check the remainder. If the remainder is zero, then x = -1 is a rational zero. If the remainder is not zero, move on to the next possible rational zero.

Step 6: The rational zeros of the polynomial function h(x) = x^3 - 4x^2 + 5x - 2 are x = 1 and x = -1.