Factor simple trinomials with leading coefficient one

Factoring a monic quadratic trinomial $x^2+bx+c$ means recognizing it as a product $(x+m)(x+n)$ , where the integers $m$ and $n$ have sum $b$ and product $c$ , including appropriate sign choices when $c$ is positive or negative. The factorization expresses the relationship between multiplication and addition and supports solving quadratic equations by the zero-product property; non-integer factors, non-monic trinomials, and more general methods are outside this scope.

Detailed Explanation: Factor simple trinomials with leading coefficient one

To factor a trinomial of the form

x^2+bx+c,

look for two integers, $m$ and $n$ , such that:

$m+n=b$ (their sum is the coefficient of $x$ )
$mn=c$ (their product is the constant)

Then write:

x^2+bx+c=(x+m)(x+n).

Example

Factor:

x^2+5x+6

Step 1: Identify $b$ and $c$ .

In $x^2+5x+6$ :

b=5,\qquad c=6

Step 2: Find two numbers whose product is $6$ and whose sum is $5$ .

The factor pairs of $6$ are:

1\text{ and }6,\qquad 2\text{ and }3

Since $2+3=5$ and $2\cdot3=6$ , the numbers are $2$ and $3$ .

Step 3: Write the factors.

Place the numbers in two binomials:

x^2+5x+6=(x+2)(x+3)

Step 4: Check by multiplying.

(x+2)(x+3) =x^2+3x+2x+6 =x^2+5x+6

Therefore,

\boxed{x^2+5x+6=(x+2)(x+3)}

When the constant $c$ is negative, the two numbers must have opposite signs. When $c$ is positive, they have the same sign; choose the signs that make their sum equal $b$ .

Learn by doing: Factor simple trinomials with leading coefficient one

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Algebraic Functions - Factor the Quadratic Equation

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