So how do you write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros with is problem: 5 and 4 + √2
irrational zeroes occur in conjugate pairs, so 4-√2 is also a root
(x-(4+√2))(x-(4-√2))(x-5)
= ((x-4)-√2)((x-4)+√2)(x-5)
= ((x-4)^2 - 2)(x-5)
To write a polynomial function with rational coefficients, a leading coefficient of 1, and the given zeros, you can use the concept of conjugate pairs for irrational zeros.
Given that the zeros are 5 and 4 + √2, we know that the corresponding conjugate of 4 + √2 will also be a zero. The conjugate of 4 + √2 is 4 - √2.
To find the polynomial function, you can multiply the factors corresponding to each zero. In this case, the factors are x - 5, x - (4 + √2), and x - (4 - √2).
Multiplying these factors together, we get:
(x - 5)(x - (4 + √2))(x - (4 - √2))
Expanding this expression will give us the polynomial function:
(x - 5)(x - 4 - √2)(x - 4 + √2)
To simplify this, we can use the difference of squares formula (a - b)(a + b) = a^2 - b^2:
(x - 5)((x - 4)^2 - (√2)^2)
(x - 5)(x^2 - 8x + 16 - 2)
(x - 5)(x^2 - 8x + 14)
Expanding this expression further will give us the final polynomial function:
f(x) = x^3 - 13x^2 + 74x - 70
To write a polynomial function, we need its zeros, which are the values of x for which f(x) equals zero. From your question, the given zeros are 5 and 4 + √2.
Since all the coefficients need to be rational and the leading coefficient is 1, we know that the polynomial can be written in factored form as:
f(x) = (x - 5)(x - (4 + √2))(x - (4 - √2))
To simplify this expression, we'll need to multiply it out and combine like terms. Let's start by expanding the factors:
(x - 5)(x - (4 + √2))(x - (4 - √2))
First, let's simplify the expression inside the second parenthesis:
(x - 5)(x - 4 - √2)(x - 4 + √2)
Now, we'll apply the distributive property to multiply the first two binomials:
(x² - 5x - 4x + 20 - √2x + 5√2 + 4√2 - 20√2)(x - 4 + √2)
Simplifying further, we can combine like terms:
(x² - 9x + 20 + √2(9 - x) + 9√2 - 16√2)(x - 4 + √2)
Finally, we can simplify the expression again, combining like terms and distributing:
(x² - 9x + 20 - x√2 + 9√2 - 16√2 + x√2 + 16 - 4x + 4√2 - 4)(x - 4 + √2)
Simplifying once more, we obtain:
(x² - 13x + 36)(x - 4 + √2)
Therefore, the polynomial function f(x) of least degree with rational coefficients, a leading coefficient of 1, and zeros 5 and 4 + √2 is:
f(x) = (x² - 13x + 36)(x - 4 + √2)