❖ Synthetic Division - A Shortcut for Long Division! ❖

What people said

Recommends Amateur Math Guy · 2013-02 · Mathematics Stack Exchange · on “Factoring Cubic Equations”

Thanks! This post along with the following videos: 1 [LINK] , 2 [LINK] , 3 [this video] , made this concept crystal clear. I appreciate the reference to the rational roots theorem which I see will be useful in the future

Recommends user404127 · 2017-01 · Mathematics Stack Exchange · on “Finding quotient and remainder”

When you factor polynomials like this over, you should end up with \begin{equation*} f(x)=p(x)(x+2)+r(x) \end{equation*} Where $r(x)$ is a polynomial with degree less than $deg(x+2)$. In this case that gives you a constant. This allows you to plug in $-2$ for $x$ to discover the remainder! \begin{equation*} f(-2)=-49 \end{equation*} It looks like you need to tweak your remainder. Have you heard of synthetic division? This video might help: [this video]

Recommends K Split X · 2017-04 · Mathematics Stack Exchange · on “How to solve polynomial of degree 4?”

We have $$x^4+x^3+x=3\to x^4+x^3+x-3=0$$ $$\text{When we have powers of $x>3$, we usually find an easy solution by testing a few values.}$$ $$\text{For example, if we plug in $x=1$, we have $1+1+1-3=0$}$$ $$\text{This means that $x=1$ is one of the solutions, and that $(x-1)$ is factor.}$$ Now we could use synthetic division to further reduce our polynomial. [this video] That will give us $x^3+2x^2+2x+3=0$ This is irrational roots, and so we cannot do much after this. Either you must use a calculator, use methods like Newton's method, or analyse the derivative and do it. There is no easy way after this. $x=-1.81,1$

Recommends Gnumbertester · 2019-01 · Mathematics Stack Exchange · on “Solution to quadratic and cubic equation with partial root”

Your answer is incorrect and there is a much simpler way of finding the roots. Since you are given a root of this cubic, you can use synthetic division to reduce it to a quadratic. You may already know how to do synthetic division, however if you don't, you can see how to do it here [this video] . Once you divide $x^3+8x^2+16x+3$ by the linear factor you are given, you are left with $x^2+5x+1$ . To find the other two roots, use the quadratic formula. Addendum: If you are familiar with traditional, polynomial long division, that works too.