Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 3 x^{2}-12 x-36 $$

Short answer

3(x - 6)(x + 2)

Step-by-step solution

Identify the Greatest Common Factor (GCF)

Examine the polynomial \(3x^2 - 12x - 36\). Identify the greatest common factor of the coefficients 3, -12, and -36. The GCF is 3.

Factor Out the GCF

Factor 3 out of each term in the polynomial: \(3(x^2 - 4x - 12)\).

Factor the Quadratic Expression

We need to factor \(x^2 - 4x - 12\). To do this, find two numbers that multiply to -12 and add to -4. The numbers -6 and 2 work: \((x - 6)(x + 2)\).

Combine All Factors

Put everything together to get the completely factored form: \(3(x - 6)(x + 2)\).

Key concepts

greatest common factor

When solving a polynomial, the first step is to find the greatest common factor (GCF). The GCF is the largest number that divides all coefficients in the polynomial without leaving a remainder.
To find the GCF of \(3x^2 - 12x - 36\), we look at the coefficients: 3, -12, and -36.
The greatest common factor of these numbers is 3.
Once identified, you can factor out the GCF from each term in the polynomial.
For our example, we factor the 3 out, transforming \(3x^2 - 12x - 36\) into \(3(x^2 - 4x - 12)\).
This simplifies the polynomial and prepares it for further factoring.

factoring quadratic expressions

After factoring out the GCF, the next step involves factoring the quadratic expression.
Quadratic expressions are polynomials of the form \(ax^2 + bx + c\).
In our example, we focus on \(x^2 - 4x - 12\).
Factoring quadratics involves finding two numbers that multiply to give the constant term (in this case -12) and add up to give the coefficient of the middle term (in this case -4).
The numbers -6 and 2 fit this requirement because \(-6 \times 2 = -12\) and \(-6 + 2 = -4\).
Using these numbers, we can rewrite the quadratic as \((x - 6)(x + 2)\).

polynomial factorization

Combining the steps of factoring out the GCF and factoring the quadratic expression gives us the fully factored polynomial.
After extracting the GCF 3 from \(3x^2 - 12x - 36\), we worked on simplifying the expression \(x^2 - 4x - 12\) into \((x - 6)(x + 2)\).
We then put it all together.
Hence, the completely factored form of the given polynomial is \(3(x - 6)(x + 2)\).
Polynomial factorization simplifies the problem and helps in finding the roots of the equation easily.
This can be useful for graphing functions and solving higher-level algebra problems.