Factoring a quadratic polynomial into two binomials can be achieved using a system of equations. This system is derived from the quadratic formula and involves finding two numbers that sum up to the coefficient of the linear term and multiply to the constant term of the quadratic polynomial.

### Key Facts

- Factoring a quadratic polynomial can be done using a system of equations. The system typically involves finding two numbers that add up to the coefficient of the linear term and multiply to the constant term of the quadratic polynomial.
- The system of equations for factoring a quadratic polynomial can be derived from the quadratic formula. By solving the system, you can find the factors of the quadratic polynomial.
- Factoring completely is a more complex process that involves combining different factoring techniques. It typically includes factoring out a greatest common factor, factoring trinomials, and factoring the difference between two squares.
- Factoring completely is a three-step process:

a. Factor out the greatest common factor, if possible.

b. Factor trinomials, if possible.

c. Factor the difference between two squares as many times as possible. - Factoring completely may require multiple applications of the difference between two squares method. This means that you may need to factor the difference between two squares more than once to fully factor a polynomial.

## Factoring Completely

Factoring completely is a more intricate process that combines multiple factoring techniques. It typically involves the following steps:

### Step 1: Factoring Out the Greatest Common Factor

If possible, factor out the greatest common factor from the polynomial.

### Step 2: Factoring Trinomials

If possible, factor the polynomial as a trinomial.

### Step 3: Factoring the Difference Between Two Squares

Repeatedly factor the difference between two squares as many times as possible.

## Example

Consider the polynomial $12x^4 – 3x^2 – 54$.

### Step 1: Factoring Out the Greatest Common Factor

The greatest common factor is $3$, so we factor it out:

12×4−3×2−54=3(4×4−x2−18)12x^4 – 3x^2 – 54 = 3(4x^4 – x^2 – 18)

### Step 2: Factoring Trinomials

3(4×2−9)(x2+2)3(4x^2 – 9)(x^2 + 2)

### Step 3: Factoring the Difference Between Two Squares

3(2x+3)(2x−3)(x2+2)3(2x + 3)(2x – 3)(x^2 + 2)

Therefore, the polynomial is factored completely as $3(2x + 3)(2x – 3)(x^2 + 2)$.

### Conclusion

Factoring polynomials using systems of equations is a viable method for quadratic polynomials. Factoring completely involves combining different factoring techniques, including factoring out the greatest common factor, factoring trinomials, and factoring the difference between two squares. This process may require multiple applications of the difference between two squares method to fully factor a polynomial.

## Sources

- Is it possible to use system of equations when factoring trinomials?
- Factoring Completely Lessons
- Example 1: Solving a Quadratic Equation by Factoring

## FAQs

### What is factoring a polynomial into systems of equations?

Factoring a polynomial into systems of equations involves finding two numbers that add up to the coefficient of the linear term and multiply to the constant term of the quadratic polynomial. These numbers can then be used to create a system of equations that can be solved to find the factors of the polynomial.

### Why would I want to factor a polynomial into systems of equations?

Factoring a polynomial into systems of equations can be useful for solving quadratic equations. By finding the factors of the quadratic, you can use the zero product property to solve for the roots of the equation.

### What are the steps involved in factoring a polynomial into systems of equations?

The steps involved in factoring a polynomial into systems of equations are:

### Set the polynomial equal to zero.

### Factor the polynomial into two binomials.

### Set each binomial equal to zero.

### Solve each equation for the variable.

### The solutions to the equations are the factors of the polynomial.

### Can I factor any polynomial into systems of equations?

No, you can only factor quadratic polynomials into systems of equations.

### What is the difference between factoring completely and factoring into systems of equations?

Factoring completely involves factoring out the greatest common factor, factoring trinomials, and factoring the difference between two squares. Factoring into systems of equations is a specific method for factoring quadratic polynomials.

### When should I use factoring into systems of equations instead of other factoring methods?

Factoring into systems of equations is most useful when you need to solve a quadratic equation.

### Are there any online resources that can help me learn more about factoring polynomials into systems of equations?

Yes, there are many online resources that can help you learn more about factoring polynomials into systems of equations. Some helpful resources include:

– Khan Academy: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratics-solve-factoring/v/example-1-solving-a-quadratic-equation-by-factoring

– Wyzant: https://www.wyzant.com/resources/lessons/math/algebra/factoring-completely/

– Mathway: https://www.mathway.com/algebra/Quadratic-Equations/Factoring-Quadratic-Equations

### Can I use a calculator to factor a polynomial into systems of equations?

Yes, you can use a calculator to factor a polynomial into systems of equations. However, it is important to understand the steps involved in factoring so that you can check your calculator’s answer.