How to write polynomials in factored form?

Steps to Solve

1. Identify common factors

Examine the polynomial for any common factors in all terms. If present, factor them out.

For example, in the polynomial $6x^2 + 9x$, the common factor is $3x$:

$$6x^2 + 9x = 3x(2x + 3)$$

2. Factor by grouping

If the polynomial has four terms, try grouping terms to factor by grouping.

For example, in the polynomial $x^3 + 3x^2 + x + 3$, group the terms as $(x^3 + 3x^2) + (x + 3)$, and factor common factors from each group:

$$ x^2(x + 3) + 1(x + 3) = (x^2 + 1)(x + 3)$$

3. Factor quadratics using the quadratic formula

For polynomials of the form $ax^2 + bx + c$, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the roots and write the polynomial as a product of linear factors.

For example, for $x^2 + 5x + 6$, the quadratic formula gives roots $x = -2$ and $x = -3$:

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

4. Use special factoring formulas

Utilize special factoring formulas like difference of squares, sum and difference of cubes:

• Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
• Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

For example, $x^2 - 9 = (x - 3)(x + 3)$

5. Check your work

Expand the factored form to ensure it simplifies back to the original polynomial.

For example, checking $x^2 + 5x + 6 = (x + 2)(x + 3)$ by expanding:

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