Factor trinomials with a non-unit leading coefficient

A quadratic trinomial $ax^2+bx+c$ , with integer coefficients and $a\neq 1$ , can be factored over the integers by relating the product $ac$ to the sum $b$ , then expressing the middle term so the resulting four terms can be grouped into two binomials. This reasoning accounts for positive and negative signs, guards against factoring $a$ and $c$ independently without checking the middle coefficient, and supports solving quadratic equations; the scope excludes irreducible trinomials and more general symbolic or non-integer factorization.

Detailed Explanation: Factor trinomials with a non-unit leading coefficient

To factor a trinomial with a leading coefficient other than $1$ , use the product $ac$ and the middle coefficient $b$ .

For a trinomial

ax^2+bx+c,

Multiply $a\cdot c$ .
Find two integers whose product is $ac$ and whose sum is $b$ .
Rewrite the middle term using those two numbers.
Group the four terms and factor each group.
Check by multiplying the binomials.

Example: Factor

6x^2+11x+3.

Here, $a=6$ , $b=11$ , and $c=3$ .

First multiply the first and last coefficients:

ac=6\cdot 3=18.

Now find two numbers that multiply to $18$ and add to $11$ . The numbers are $2$ and $9$ :

2\cdot 9=18 \qquad\text{and}\qquad 2+9=11.

Use $2x$ and $9x$ to split the middle term:

6x^2+11x+3 = 6x^2+2x+9x+3.

Group the terms:

(6x^2+2x)+(9x+3).

Factor each group:

2x(3x+1)+3(3x+1).

Both groups contain the factor $(3x+1)$ , so factor it out:

(2x+3)(3x+1).

Therefore,

\boxed{6x^2+11x+3=(2x+3)(3x+1)}.

Check by multiplying:

(2x+3)(3x+1) =6x^2+2x+9x+3 =6x^2+11x+3.

The signs matter: the two numbers must have the correct product and the correct sum. Factoring the first and last terms separately without checking the middle coefficient may produce incorrect factors.

Learn by doing: Factor trinomials with a non-unit leading coefficient

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Factor the Quadratic Equation with Coefficient - Standard Form To Answer

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Factor trinomials with a non-unit leading coefficient

Detailed Explanation: Factor trinomials with a non-unit leading coefficient

Learn by doing: Factor trinomials with a non-unit leading coefficient

Practice with unlimited practice problems

Practice:

Factor the Quadratic Equation with Coefficient - Standard Form To Answer