Now let's move on to factorizations that may require two or...

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility.
$$3 r^{3}-27 r^{2}-210 r$$

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Key Concepts

Factoring Quadratics

Once the GCF is factored out, any resulting quadratic expression can often be factored into two linear factors. This process involves finding two numbers that multiply to the product of the quadratic’s coefficient and constant term while adding to the coefficient of the middle term. Factoring quadratics is a fundamental technique for solving quadratic equations and simplifying polynomial expressions.

Polynomial Factorization Techniques

This overarching concept involves applying multiple methods in sequence to break down a polynomial into simpler factors. Techniques such as extracting the GCF and factoring quadratics are common steps in factoring more complicated polynomials. Mastery of these techniques is crucial for solving and simplifying polynomial equations efficiently.

Greatest Common Factor

In polynomial factorization, the greatest common factor (GCF) is the largest expression that divides each term of the polynomial. By factoring out the GCF, the expression is simplified, which often makes subsequent steps—such as factoring a quadratic—more straightforward. This is a preliminary step in many factoring problems.

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