Factoring and Recognizing Structure

Definition To factor an expression means to write it as an equivalent product of two or more expressions. Each expression in that product is a factor of the original. For polynomials with integer coefficients, factoring usually means writing the polynomial as a product of polynomials of lower degree, again with integer (or at least real) coefficients.

Factoring is a rewriting step. It produces an equivalent expression and does not by itself solve anything. Only when the factored expression is set equal to $0$ does it lead to solutions.

Definition The greatest common factor (GCF) of the terms of a polynomial is the product of the greatest common factor of the coefficients and, for each variable, the lowest power of that variable appearing in every term. By the distributive property,

so the GCF can be pulled out in front, leaving the quotient of each term by the GCF inside the parentheses.

Definition A quadratic trinomial is a polynomial $ax^2 + bx + c$ with $a \ne 0$, usually with $b \ne 0$ and $c \ne 0$ so that it has three terms. If $a = 1$ and there are numbers $p$ and $q$ with $p + q = b$ and $pq = c$, then

If $a \ne 1$, look for numbers $p$ and $q$ with $pq = ac$ and $p + q = b$, split the middle term as $px + qx$, and factor by grouping. Not every trinomial factors with integer coefficients.

Definition For all real numbers $a$ and $b$,

A sum of squares behaves differently. If $k > 0$ and $c > 0$, then $kx^2 + c$ is positive for every real $x$, so it has no real zero. A linear factor $px + q$ with real $p \ne 0$ would be zero at $x = -\frac{q}{p}$, so $kx^2 + c$ has no linear factor with real coefficients and does not factor over the real numbers.

Definition For all real numbers $a$ and $b$,

A trinomial is a perfect square exactly when its first and last terms are squares $a^2$ and $b^2$ and its middle term is $2ab$ or $-2ab$.

Definition Factoring by grouping splits the terms of a polynomial into groups, factors each group so that the same expression $B$ appears in every group, and then pulls out $B$ with the distributive property,

It works only when the groups really do share the same factor, which sometimes requires pulling out a negative number from a group.

Definition An expression has nested structure when a whole subexpression plays the role of a single variable. If an expression can be written as $P(u)$, where $P$ is a polynomial and $u$ stands for a subexpression such as $x + 1$ or $x^2$, then any factorization of $P(u)$ gives a factorization of the original expression after $u$ is replaced by that subexpression.