To factor a quadratic equation of the form $ax^2 + bx + c = 0$, follow these steps:

1. Check if the Quadratic is Factorable

First, determine if the quadratic can be factored over the integers by calculating the discriminant ($b^2 - 4ac$). If the discriminant is a perfect square, then the quadratic can be factored over the integers.

2. Factorisation by Splitting the Middle Term

For a quadratic equation $ax^2 + bx + c$, follow these steps to factor it by splitting the middle term:

1. Multiply $a$ and $c$: Multiply the coefficient of $x^2$ (which is $a$) and the constant term (which is $c$).

2. Find Two Numbers that Multiply to $ac$ and Add to $b$: Identify two numbers $m$ and $n$ such that $m \times n = ac$ and $m + n = b$.

3. Rewrite the Middle Term: Rewrite the quadratic equation, splitting the middle term $bx$ into two terms using the numbers $m$ and $n$.

4. Factor by Grouping: Group the terms into two pairs and factor out the common factors from each pair.

5. Factor Out the Common Binomial Factor: Factor out the common binomial factor to get the factored form.

Example

Let’s factor the quadratic equation $2x^2 + 5x + 3$.

1. Multiply $a$ and $c$: $$ a = 2, \quad b = 5, \quad c = 3 $$ $$ ac = 2 \times 3 = 6 $$

2. Find Two Numbers: We need two numbers that multiply to 6 and add to 5. The numbers are 2 and 3: $$ m = 2, \quad n = 3 $$

3. Rewrite the Middle Term: Rewrite $5x$ as $2x + 3x$: $$ 2x^2 + 2x + 3x + 3 $$

4. Factor by Grouping: Group the terms and factor out the common factors from each pair: $$ 2x(x + 1) + 3(x + 1) $$

5. Factor Out the Common Binomial Factor: Factor out $(x + 1)$: $$ (2x + 3)(x + 1) $$

Therefore, the quadratic equation $2x^2 + 5x + 3$ factors to $(2x + 3)(x + 1)$.

To factor a cubic polynomial with only a first term and a constant, such as $ax^3 + b$, you can follow these steps:

Example: Factor $2x^3 - 16$

1. Identify the Greatest Common Factor (GCF): First, identify any common factors between the terms. In this case, the GCF of $2x^3$ and $-16$ is $2$.

   $$ 2x^3 - 16 = 2(x^3 - 8) $$

2. Recognize the Difference of Cubes: The expression inside the parentheses, $x^3 - 8$, is a difference of cubes. The difference of cubes can be factored using the formula:

   $$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$

   Here, $a = x$ and $b = 2$ because $8 = 2^3$.

3. Apply the Difference of Cubes Formula: Substitute $a = x$ and $b = 2$ into the formula:

   $$ x^3 - 2^3 = (x - 2)(x^2 + 2x + 4) $$

4. Combine with the GCF: Now, combine this result with the GCF factored out earlier:

   $$ 2(x^3 - 8) = 2(x - 2)(x^2 + 2x + 4) $$

Therefore, the factored form of $2x^3 - 16$ is:

$$ 2(x - 2)(x^2 + 2x + 4) $$

Steps for Factoring a General Cubic Polynomial $ax^3 + b$:

1. Factor out the GCF: $$ ax^3 + b = a(x^3 + \frac{b}{a}) $$

2. Recognize the Difference of Cubes: Rewrite the expression inside the parentheses as a difference of cubes if possible: $$ x^3 + \frac{b}{a} = x^3 - (-\frac{b}{a}) $$

3. Apply the Difference of Cubes Formula: $$ x^3 - (-\frac{b}{a}) = (x - \sqrt[3]{-\frac{b}{a}})(x^2 + x\sqrt[3]{-\frac{b}{a}} + (\sqrt[3]{-\frac{b}{a}})^2) $$

4. Combine with the GCF: Multiply the factored form by the GCF to get the final factored polynomial.