Under which conditions ( as to the nature of coefficients) a quadratic trinomial can be factored in this way?

Question

Suppose I want to factor $P(x)=5x^2 -17x +6$.

• Set $5x^2 -17x +6 = 0 $

$5x^2 -17x +6 = 0 $

$\rightarrow 5x^2 -17x = -6$

$\rightarrow x(5x-17) = -6$

$\rightarrow x(5x-17) = -6$ $= (-2)(3) = (3) (-2) = (6)(-1) = (-1)(6)...$

• By trial and error, find a solution : $x = 3$ ( since $3 ( (5\times3) - 17) = -6$). That is number $3$ is one value of $x$ that brings one of the desired pairs of factors, namely , the product : $(3)(-2)$.

• So, by the factor theorem $(x-3)$ must be a factor of the the original quadratic trinomial.

• So $P(x)= (x-3) Q(x)$ where $Q(x)$ is a polynomial of degree $(2-1) = 1$, hence linear. That is $ Q(x) = (ax+b)$.

• Find coefficients $a$ and $b$ such that : $(x-3) ( ax+b) = P(x)=5x^2 -17x +6$.

• If $(x-3) ( ax+b) = 5x^2 -17x +6$

then ( by developping the LHS) $ax^2= 5x^2$ implying $a = 5$ and $(-3)b=6$, implying $b=-2$.

• Hence : $Q(x)= ( 5x -2)$ and $P(x)= (x-3) ( 5x-2)$.

Answer 1

Obiusvly, you can factor with zuch method only the solutions belongs to $Z$ or $Q$. In yahat case is very simple to find a possible zero by trial and error. At the contrary, if you let $P(x)=x^2-4x+2$, you can't apply your method. Also, I advise you to check the Rational Root theorem.

Answer 2

This is the Rational Root Theorem. It says that if the coefficients of a polynomial are integers, then any rational root can be written as $p/q$ where $p$ is an integer factor of the constant term, and $q$ is an integer factor of the highest order term. So in your case $p$ can be $\pm1, \pm2, \pm 3,\pm 6$ and $q$ can be $\pm1, \pm5$

Answer 3

I'm not entirely sure what you're asking OR what you're doing but the way I was taught to factor a quadratic trinomial is to find two numbers, which we will call $a$ and $b$, such that $ab$ is equal to the first coefficient (on the $x^2$ term) and $a+b$ is equal to the middle coefficient (on the $x$ term). You then use these numbers to "split up" the $x$ term) and apply the "two-two grouping" method. Finally, you factor out the GCF of the remaining two terms and you're done. Sorry if this isn't what you were asking for, just trying to help out.