Form a polynomial whose zeros and degree are given

Form a polynomial whose zeros and degree are given. Zeros: -1,1,2; degree: 3 Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. f(x) = (Simplify your answer.). Form a polynomial whose real zeros and degree are given. Zeros: -3, -2,1,2; degree: 4 Type a polynomial with integer coefficients and a leading coefficient of 1. f(x) = (Simplify your answer.). Form a polynomial whose zeros and degree are given. Zeros: -1, multiplicity 1; 4, multiplicity 2; degree 3 Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. f(x) = (Simplify your answer.).

The Correct Answer and Explanation is :

Problem 1: Zeros: -1, 1, 2; Degree: 3

To form a polynomial from the given zeros, we use the fact that if a number ( r ) is a zero of a polynomial, then ( (x – r) ) is a factor of the polynomial.

Given zeros: ( -1, 1, 2 ), the corresponding factors are:

• ( (x + 1) ) for zero -1
• ( (x – 1) ) for zero 1
• ( (x – 2) ) for zero 2

Thus, the polynomial is:

[
f(x) = (x + 1)(x – 1)(x – 2)
]

Next, expand this expression step by step.

1. First, expand ( (x + 1)(x – 1) ) using the difference of squares formula:
   [
   (x + 1)(x – 1) = x^2 – 1
   ]
2. Now multiply ( (x^2 – 1) ) by ( (x – 2) ):
   [
   (x^2 – 1)(x – 2) = x^3 – 2x^2 – x + 2
   ]

So the final polynomial is:
[
f(x) = x^3 – 2x^2 – x + 2
]

Problem 2: Zeros: -3, -2, 1, 2; Degree: 4

Given zeros: ( -3, -2, 1, 2 ), the corresponding factors are:

• ( (x + 3) ) for zero -3
• ( (x + 2) ) for zero -2
• ( (x – 1) ) for zero 1
• ( (x – 2) ) for zero 2

Thus, the polynomial is:

[
f(x) = (x + 3)(x + 2)(x – 1)(x – 2)
]

To expand this, we proceed step by step.

1. First, expand ( (x + 3)(x + 2) ):
   [
   (x + 3)(x + 2) = x^2 + 5x + 6
   ]
2. Next, expand ( (x – 1)(x – 2) ):
   [
   (x – 1)(x – 2) = x^2 – 3x + 2
   ]
3. Now multiply the two quadratics ( (x^2 + 5x + 6) ) and ( (x^2 – 3x + 2) ):
   [
   (x^2 + 5x + 6)(x^2 – 3x + 2) = x^4 + 2x^3 – 7x^2 + 4x + 12
   ]

Thus, the polynomial is:
[
f(x) = x^4 + 2x^3 – 7x^2 + 4x + 12
]

Problem 3: Zeros: -1 (multiplicity 1), 4 (multiplicity 2); Degree: 3

Given zeros: ( -1 ) with multiplicity 1 and ( 4 ) with multiplicity 2, the corresponding factors are:

• ( (x + 1) ) for zero -1
• ( (x – 4)^2 ) for zero 4 with multiplicity 2

Thus, the polynomial is:

[
f(x) = (x + 1)(x – 4)^2
]

To expand this expression:

1. First, expand ( (x – 4)^2 ):
   [
   (x – 4)^2 = x^2 – 8x + 16
   ]
2. Now multiply ( (x + 1) ) by ( (x^2 – 8x + 16) ):
   [
   (x + 1)(x^2 – 8x + 16) = x^3 – 8x^2 + 16x + x^2 – 8x + 16
   ]
3. Combine like terms:
   [
   f(x) = x^3 – 7x^2 + 8x + 16
   ]

Summary of Polynomials:

1. ( f(x) = x^3 – 2x^2 – x + 2 )
2. ( f(x) = x^4 + 2x^3 – 7x^2 + 4x + 12 )
3. ( f(x) = x^3 – 7x^2 + 8x + 16 )

Visualization of the Polynomials:

I’ll generate a graph that shows the roots and behavior of the polynomials.

Here is a visual representation of the three polynomials based on their zeros:

• The blue curve represents ( f(x) = x^3 – 2x^2 – x + 2 ) with zeros at -1, 1, and 2.
• The red curve represents ( f(x) = x^4 + 2x^3 – 7x^2 + 4x + 12 ) with zeros at -3, -2, 1, and 2.
• The green curve represents ( f(x) = x^3 – 7x^2 + 8x + 16 ) with zeros at -1 (multiplicity 1) and 4 (multiplicity 2).

Each polynomial is plotted along with its zeros clearly marked, showing how the degree and multiplicity affect the shape of the curves. Let me know if you’d like more clarification!