Question

Write the equation of the lowest degree polynomial with solutions x = -1, 1, 2 +- root5

Answer 1

Given the roots of the equation to be:

`x=-1,1,2\pm\sqrt[]{5}`

Therefore, the factors are:

`(x+1),(x-1),(x-2+\sqrt[]{5}),(x-2-\sqrt[]{5})`

To get the polynomial, we will multiply the factors together:

`\Rightarrow(x+1)(x-1)(x-2+\sqrt[]{5})(x-2-\sqrt[]{5})`

Expanding in pairs, we can have:

`\Rightarrow\lbrack(x+1)(x-1)\rbrack(x-2+\sqrt[]{5})(x-2-\sqrt[]{5})`

Recall the Difference of Two Squares given to be:

`a^2-b^2=(a-b)(a+b)`

Hence, the first pair becomes:

`(x+1)(x-1)=x^2-1^2=x^2-1`

Hence, the polynomial becomes:

`\Rightarrow(x^2-1)\lbrack(x-2+\sqrt[]{5})(x-2-\sqrt[]{5})\rbrack`

Expanding the pair:

`(x-2+\sqrt[]{5})(x-2-\sqrt[]{5})=x^2-2x-x\sqrt[]{5}-2x+4+2\sqrt[]{5}+x\sqrt[]{5}-2\sqrt[]{5}-5=x^2-4x-1`

Hence, the expression becomes:

`\Rightarrow(x^2-1)(x^2-4x-1)`

Expanding, we get:

`\Rightarrow x^4-4x^3-x^2-x^2+4x+1=x^4-4x^3-2x^2+4x+1^{}`

Therefore, the polynomial is:

`\Rightarrow x^4-4x^3-2x^2+4x+1`