Using Horner's Method

Solution 1:

$6x^7-7x^5+2x^4-10x^2+20x-6$ =

$6x^7+0x^6-7x^5+2x^4+0x^3-10x^2+20x-6$

The method is essentially start with the coefficient of the highest power, multiply by $x$ and add the next coefficient. Stop when you add the constant coefficient.

So steps in the iteration go:

$6$

$6x[+0]$

$6x^2-7$

$6x^3-7x+2$

$6x^4-7x^2+2x[+0]$

$6x^5-7x^3+2x^2-10$

$6x^6-7x^4+2x^3-10x+20$

$6x^7-7x^5+2x^4-10x^2+20x-6$

Trust this helps

Solution 2:

Horner's method or form is also sometimes called nested form. You can think of it as starting with the whole polynomial $$6x^7-7x^5+2x^4-10x^2+20x-6,$$ setting aside the constant term (if it's zero, you can set aside a zero here) and factoring out an $x$ from the remaining terms $$(6x^6-7x^4+2x^3-10x+20)x-6,$$ and then repeating the process for the innermost polynomial (the part that isn't yet fully in nested form: $$((6x^5-7x^3+2x^2-10)x+20)x-6$$ $$(((6x^4-7x^2+2x)x-10)x+20)x-6$$ $$((((6x^3-7x+2)x+0)x-10)x+20)x-6$$ $$(((((6x^2-7)x+2)x+0)x-10)x+20)x-6$$ $$((((((6x)x-7)x+2)x+0)x-10)x+20)x-6$$ $$(((((((6)x+0)x-7)x+2)x+0)x-10)x+20)x-6$$

Alternately, if your polynomial is $$p(x)=\sum_{k=0}^{n}a_kx^k=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$$ you can think of $p(x)$ in nested form as $p_0$ where $p_n=a_n$ and $p_{k-1}=p_kx+a_{k-1}$. That is, start with the leading coefficient ($a_n$). Multiply by $x$ and add the next coefficient (adding zero to fill in for "missing" powers of $x$), repeating until you've added the constant term ($a_0$). Back to the same example: $$\begin{align} p_7&=6\\ p_6&=(6)x+0\\ p_5&=((6)x+0)x-7\\ p_4&=(((6)x+0)x-7)+2\\ p_3&=((((6)x+0)x-7)+2)x+0\\ p_2&=(((((6)x+0)x-7)+2)x+0)-10\\ p_1&=((((((6)x+0)x-7)+2)x+0)-10)+20\\ p_0&=(((((((6)x+0)x-7)+2)x+0)-10)+20)-6 \end{align}$$