In Exercises 65–92, factor completely, or state that the polynomial is prime. x²y−16y+32−2x²

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Group the terms of the polynomial into two pairs: $ (x^2y - 2x^2) $ and $ (-16y + 32) $.

Factor out the greatest common factor (GCF) from each group: $ x^2(y - 2) $ from the first group and $ -16(y - 2) $ from the second group.

Notice that $ (y - 2) $ is a common factor in both terms. Factor $ (y - 2) $ out of the entire expression: $ (y - 2)(x^2 - 16) $.

Recognize that $ x^2 - 16 $ is a difference of squares. Apply the difference of squares formula $ a^2 - b^2 = (a - b)(a + b) $ to factor $ x^2 - 16 $ as $ (x - 4)(x + 4) $.

Combine all the factors to write the completely factored form of the polynomial: $ (y - 2)(x - 4)(x + 4) $.

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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 rewriting a polynomial expression as a product of its factors. This process is essential for simplifying expressions and solving equations. Common techniques include factoring out the greatest common factor, using the difference of squares, and applying the quadratic formula when necessary.

Identifying Prime Polynomials

A polynomial is considered prime if it cannot be factored into simpler polynomials with integer coefficients. Recognizing prime polynomials is crucial in algebra, as it helps determine the limits of simplification and the methods needed for solving polynomial equations.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. This step is often necessary before factoring, as it helps to organize the polynomial into a standard form, making it easier to identify potential factors.

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