Theory

Factorization, Ruffini's rule and polynomial division

What a polynomial is, how special products are expanded and recognized, how polynomial division is carried out and how a polynomial is factored with Ruffini's rule.

On this page

Polynomials and degree
Special products

Polynomials and degree

1. Polynomial and degree

A polynomial in the variable $x$ is a sum of monomials,

P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0},

with $a_{n} \neq = 0$ . The numbers $a_{n}, \dots, a_{0}$ are the coefficients, $a_{n}$ is the leading coefficient, $a_{0}$ is the constant term and $n$ is the degree, denoted by $de g P$ . A number $c$ is a root of $P$ if $P (c) = 0$ . The polynomial $P (x) = 2 x^{3} - 5 x + 1$ has degree $3$ , leading coefficient $2$ and constant term $1$ ; the coefficient of $x^{2}$ is $0$ .

2. Identity principle for polynomials

Two polynomials are equal if and only if they have the same degree and the same coefficients, term by term. The principle makes it possible to determine unknown coefficients by equating the coefficients of terms of the same degree, and it is used to check divisions and factorizations. From the equality $x^{2} + b x + c = (x - 1) (x - 3) = x^{2} - 4 x + 3$ it follows that $b = - 4$ and $c = 3$ .

Remark

In the product of two polynomials the degrees add up, so $de g (P Q) = de g P + de g Q$ . For this reason a polynomial of degree $4$ factored into two factors of positive degree has factors of degrees $1$ and $3$ or $2$ and $2$ .

Special products

The following identities hold for all values of $A$ and $B$ and are used in both directions, to expand and to factor.

1. Square and cube of a binomial

(A + B)^{2} = A^{2} + 2 A B + B^{2}, (A - B)^{2} = A^{2} - 2 A B + B^{2}, (A + B)^{3} = A^{3} + 3 A^{2} B + 3 A B^{2} + B^{3}, (A - B)^{3} = A^{3} - 3 A^{2} B + 3 A B^{2} - B^{3} .

Read from right to left, they factor: a trinomial is the square of a binomial when two terms are squares and the third is twice the product of the two bases, and the sign of this double product determines the sign in the binomial.

Example

$(2 x + 3)^{2} = 4 x^{2} + 12 x + 9$ . In the opposite direction, $x^{2} - 10 x + 25 = (x - 5)^{2}$ , since $x^{2}$ and $25$ are the squares of $x$ and $5$ and $- 10 x$ is the double product $- 2 \cdot x \cdot 5$ . For the cube, $(x + 2)^{3} = x^{3} + 6 x^{2} + 12 x + 8$ .

2. Difference of squares

A^{2} - B^{2} = (A - B) (A + B)

It is the quickest factorization to recognize: two squares separated by a minus sign. It can also be applied several times in a row, when one of the factors obtained is itself a difference of squares.

Example

$9 x^{2} - 16 = (3 x - 4) (3 x + 4)$ . With two successive applications, $x^{4} - 1 = (x^{2} - 1) (x^{2} + 1) = (x - 1) (x + 1) (x^{2} + 1)$ .

Warning

The sum of two squares $A^{2} + B^{2}$ cannot be factored into first-degree factors with real coefficients. For example $x^{2} + 4 \geq 4 > 0$ for every $x$ , so $x^{2} + 4$ has no real roots and no factor of the form $x - c$ divides it.

3. Sum and difference of cubes

A^{3} + B^{3} = (A + B) (A^{2} - A B + B^{2}), A^{3} - B^{3} = (A - B) (A^{2} + A B + B^{2})

The second factor, called a false square, has a negative discriminant and no real roots, so the factorization does not continue; the discriminant criterion is found in the chapter on polynomial equations and inequalities.

Example

$x^{3} + 8 = (x + 2) (x^{2} - 2 x + 4)$ , with $A = x$ and $B = 2$ . Similarly $27 x^{3} - 1 = (3 x - 1) (9 x^{2} + 3 x + 1)$ , with $A = 3 x$ and $B = 1$ .

The theory continues with 5 more sections (polynomial division, remainder theorem and ruffinis theorem, ruffinis rule…), plus all worked examples, the solutions and the PDF.

Unlock now