can someone help me write a polynomial to the 7th degree, 6 unique zeros, -1 for a leading coefficient, and a zero with a multiplicity of two?

Question

anonymous · Answer

A degree-7 polynomial will look like this:
$$a_0+a_1x+\cdots+a_6x^6+a_7x^7$$
The leading coefficient should be $-1$, which means $a_7=-1$ (power terms are usually written in descending order, I have it ascending).

If we call $r$ a zero of the polynomial, then we can write the polynomial in a factored form.
$$a_7(x-r)(b_0+b_1x+\cdots+b_5x^5+b_6x^6)$$
Suppose $r$ is a zero of multiplicity 2. Then we would have
$$a_7(x-r)^2(c_0+c_1x+\cdots+c_4x^4+c_5x^5)$$
Hopefully you can see the pattern here.

For your polynomial of degree 7, let $r_1,...r_6$ denote the 6 unique roots, and let $r_1$ be the one of multiplicity 2. Then your polynomial would be
$$a_7(x-r_1)^2(x-r_2)(x-r_3)(x-r_4)(x-r_5)(x-r_6)$$
From here, you just need to plug in distinct numbers for each root $r_i$, and $a_7=-1$.