Form a polynomial whose zeros and degree are given

1. Form a polynomial whose zeros and degree are given.
   Zeros: -3,3.4; degree: 3
   Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below.
   f(x) (Simplify your answer.)
2. Form a polynomial whose real zeros and degree are given. Zeros: -2,0,6; degree: 3
   Type a polynomial with integer coefficients and a leading coefficient of 1.
   f(x)= (Simplify your answer.)

The Correct Answer and Explanation is :

Solution to Problem 1: Zeros: -3, 3.4; Degree: 3

To form the polynomial with integer coefficients and a leading coefficient of 1, we follow these steps:

1. Zeros Given: -3 and 3.4

• The polynomial must have these values as its zeros.

1. Degree: 3

• The degree of the polynomial is 3, meaning there will be three factors in the polynomial.
• For the zero -3, the factor will be:
  [
  (x + 3)
  ]
• For the zero 3.4, the factor will be:
  [
  (x – 3.4)
  ]
  However, since we need integer coefficients, we’ll multiply this factor by 10 to eliminate the decimal, giving us:
  [
  10(x – 3.4) = 10(x – \frac{17}{5}) = 5x – 17
  ]
  So, the factor corresponding to the zero 3.4 becomes (5x – 17).

1. Third Zero: We need one more zero to make the degree 3. The simplest choice is (x = 0), corresponding to the factor:
   [
   x
   ]
2. Final Polynomial: Now we combine all the factors:
   [
   f(x) = (x + 3)(x)(5x – 17)
   ]
   Multiply these together:
   [
   f(x) = x(x + 3)(5x – 17) = x(5x^2 + 15x – 17x – 51) = x(5x^2 – 2x – 51)
   ]
   Expanding further:
   [
   f(x) = 5x^3 – 2x^2 – 51x
   ]

Thus, the polynomial is:
[
f(x) = 5x^3 – 2x^2 – 51x
]

Solution to Problem 2: Zeros: -2, 0, 6; Degree: 3

To form the polynomial with integer coefficients and a leading coefficient of 1, we follow these steps:

1. Zeros Given: -2, 0, 6

• The polynomial must have these values as its zeros.

1. Degree: 3

• The degree of the polynomial is 3, so we expect three factors in the polynomial.

1. Factors:

• For the zero -2, the factor will be:
  [
  (x + 2)
  ]
• For the zero 0, the factor will be:
  [
  x
  ]
• For the zero 6, the factor will be:
  [
  (x – 6)
  ]

1. Final Polynomial: Now we multiply all these factors:
   [
   f(x) = (x + 2)(x)(x – 6)
   ]
   First, multiply the first two factors:
   [
   (x + 2)(x) = x^2 + 2x
   ]
   Now, multiply this by the third factor:
   [
   f(x) = (x^2 + 2x)(x – 6) = x^3 – 6x^2 + 2x^2 – 12x = x^3 – 4x^2 – 12x
   ]

Thus, the polynomial is:
[
f(x) = x^3 – 4x^2 – 12x
]

Summary:

1. For the zeros -3, 3.4, and degree 3, the polynomial is:
   [
   f(x) = 5x^3 – 2x^2 – 51x
   ]
2. For the zeros -2, 0, 6, and degree 3, the polynomial is:
   [
   f(x) = x^3 – 4x^2 – 12x
   ]

Now, let’s generate the image for these polynomials.

Here are the graphs of the two polynomials:

1. The first polynomial, ( f(x) = 5x^3 – 2x^2 – 51x ), has zeros at -3, 3.4, and a degree of 3.
2. The second polynomial, ( f(x) = x^3 – 4x^2 – 12x ), has zeros at -2, 0, and 6.

These graphs illustrate the behavior of the functions based on their respective zeros. Let me know if you need further clarifications!