Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.



1) -2, -1, 1

2) i, 4

3) i, 2 - √3

4) 1, 4, 1 + √2

2) x^3 - 4x^2 + x - 4

extra credit: why a cubic, not a quadratic?

4) x^3 - (6+√2)x^2 + (9+5√2)x - 4(1+√2)
minimum degree. To get rid of the √2, add a root of 1-√2 to get
x^4 - 7x^3 + 13x^2 - 3x - 4

-12-18i

To find a polynomial function with the given zeros, follow these steps:

1) For the first problem, the zeros are -2, -1, and 1.
To find the polynomial function, we need to find the factors corresponding to each zero. Since all the zeros are given as real numbers, the factors will be (x + 2), (x + 1), and (x - 1).

2) The polynomial function can be found by multiplying all the factors:
f(x) = (x + 2)(x + 1)(x - 1).

3) For the second problem, the zeros are i and 4. Since i is not a real number, its conjugate (-i) is also a zero. So, the factors are (x - i), (x + i), and (x - 4).

4) The polynomial function can be found by multiplying all the factors:
f(x) = (x - i)(x + i)(x - 4).

5) For the third problem, the zeros are i and 2 - √3. The conjugate of 2 - √3 is 2 + √3, so the factors are (x - i) and (x - 2 + √3).

6) The polynomial function can be found by multiplying all the factors:
f(x) = (x - i)(x - 2 + √3).

7) For the fourth problem, the zeros are 1, 4, and 1 + √2. The factors are (x - 1), (x - 4), and (x - 1 - √2).

8) The polynomial function can be found by multiplying all the factors:
f(x) = (x - 1)(x - 4)(x - 1 - √2).

These are the polynomial functions of least degree with rational coefficients and a leading coefficient of 1 for each given set of zeros.

I hope this helps! Let me know if you have any further questions.

To find a polynomial function with rational coefficients that has the given zeros, follow these steps:

1) For each given zero, create a factor of the polynomial function.

2) Multiply all the factors together to obtain the polynomial function.

Let's go through each case:

1) For zeros -2, -1, and 1, the corresponding factors of the polynomial function are (x + 2), (x + 1), and (x - 1), respectively. To find the polynomial function, multiply these factors together:

f(x) = (x + 2)(x + 1)(x - 1)

Expanding this gives:
f(x) = (x² + 3x + 2)(x - 1)
= x³ + 3x² + 2x - x² - 3x - 2
= x³ + 2x² - x - 2

2) For zeros i and 4, the corresponding factors of the polynomial function are (x - i) and (x - 4), respectively. Multiply these factors together:

f(x) = (x - i)(x - 4)

Expanding this gives:
f(x) = x² - 4x - ix + 4i
= x² - 4x - ix + 4i

Since we want the polynomial to have rational coefficients, we can write the imaginary terms in terms of i as follows:
f(x) = x² - 4x - ix + 4i
= x² - 4x - i(x - 4i)

3) For zeros i and 2 - √3, the corresponding factors are (x - i) and (x - (2 - √3)), respectively. Multiply these factors together:

f(x) = (x - i)(x - (2 - √3))

Expanding this gives:
f(x) = (x - i)(x - 2 + √3)

To simplify the expression, use the conjugate: (a + b)(a - b) = a² - b²
(x - i)(x - 2 + √3) = (x - i)(x - 2 + √3)(x - 2 - √3)(x - i)

Multiplying the terms gives:
f(x) = [(x² - 2x - √3x + i√3) - (2x - 4 + 2√3 - √3x - √3i + i√3)](x - i)
= [(x² - 4x + 4) - (√3x + √3i + 2√3)](x - i)
= [x² - 4x + 4 - √3x - √3i - 2√3](x - i)
= x³ - x²(√3) + 3x² - 4x - 2√3x + 2√3i + 2√3x - 2√3i - √3x² + √3xi - 2√3x + 2√3i + 2√3
= x³ + (3 - √3)x² - (4 - √3)x + 2√3(1 - i)

4) For zeros 1, 4, and 1 + √2, the corresponding factors are (x - 1), (x - 4), and (x - (1 + √2)), respectively. Multiply these factors together:

f(x) = (x - 1)(x - 4)(x - (1 + √2))

Expanding this gives:
f(x) = (x - 1)(x - 4)(x - 1 - √2)

To simplify the expression, use the conjugate: (a + b)(a - b) = a² - b²
(x - 1)(x - 4)(x - 1 - √2) = (x - 1)(x - 4)(x - 1 + √2)(x - 4)(x - 1 - √2)

Multiplying the terms gives:
f(x) = [(x² - x + √2x - 4x + 4 - 4√2) - (x - 5 + √2 - √2x - 4 - 4√2 + √2x - √2)](x - 1)
= [(x² - 5x + 4 - 4√2) - (x - 3 -4√2)](x - 1)
= [x² - 5x + 4 - 4√2 - x + 3 + 4√2](x - 1)
= x³ - x² - 4x + 12

Therefore, the polynomial functions with rational coefficients and the given zeros are:
1) f(x) = x³ + 2x² - x - 2
2) f(x) = x² - 4x - i(x - 4i)
3) f(x) = x³ + (3 - √3)x² - (4 - √3)x + 2√3(1 - i)
4) f(x) = x³ - x² - 4x + 12