In Exercises 65–92, factor completely, or state that the polynomial is prime. x³+2x²−9x−18

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Group the terms of the polynomial into two pairs: $ (x^3 + 2x^2) $ and $ (-9x - 18) $. This is called factoring by grouping.

Factor out the greatest common factor (GCF) from each group. From $ x^3 + 2x^2 $, the GCF is $ x^2 $, so it becomes $ x^2(x + 2) $. From $ -9x - 18 $, the GCF is $ -9 $, so it becomes $ -9(x + 2) $.

Notice that both groups now contain the common factor $ (x + 2) $. Factor $ (x + 2) $ out of the entire expression, resulting in $ (x + 2)(x^2 - 9) $.

Observe that $ x^2 - 9 $ is a difference of squares. Recall the formula for factoring a difference of squares: $ a^2 - b^2 = (a - b)(a + b) $. Here, $ x^2 - 9 $ can be factored as $ (x - 3)(x + 3) $.

Combine all the factors to write the completely factored form of the polynomial: $ (x + 2)(x - 3)(x + 3) $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler polynomial factors. This process is essential for simplifying expressions and solving equations. Techniques include finding common factors, using the distributive property, and applying special factoring formulas such as the difference of squares or perfect square trinomials.

Rational Root Theorem

The Rational Root Theorem provides a method for identifying possible rational roots of a polynomial equation. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem is useful for testing potential roots to simplify the polynomial.

Synthetic Division

Synthetic division is a simplified form of polynomial long division that allows for quicker division of a polynomial by a linear factor. It is particularly useful when applying the Rational Root Theorem to test potential roots. This method reduces the polynomial's degree and helps in finding factors or confirming if the polynomial is prime.

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