Factor x^2 - 5x + 6.

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

Factor x^2 - 5x + 6.

Explanation:
To factor a quadratic with leading coefficient 1, find two numbers that multiply to the constant term and add to the middle coefficient. Here, the constant term is 6 and the middle term coefficient is -5, so we need two numbers that multiply to 6 and add to -5. Those numbers are -2 and -3. Rewriting -5x as -2x - 3x, you can factor by grouping: x^2 - 2x - 3x + 6 = x(x - 2) - 3(x - 2) = (x - 3)(x - 2). Since multiplication is commutative, this is the same as (x - 2)(x - 3). Expanding confirms: x^2 - 3x - 2x + 6 = x^2 - 5x + 6. The other pairings multiply to 6 but give the wrong sum, so they don’t produce the correct middle term when expanded.

To factor a quadratic with leading coefficient 1, find two numbers that multiply to the constant term and add to the middle coefficient. Here, the constant term is 6 and the middle term coefficient is -5, so we need two numbers that multiply to 6 and add to -5. Those numbers are -2 and -3. Rewriting -5x as -2x - 3x, you can factor by grouping: x^2 - 2x - 3x + 6 = x(x - 2) - 3(x - 2) = (x - 3)(x - 2). Since multiplication is commutative, this is the same as (x - 2)(x - 3). Expanding confirms: x^2 - 3x - 2x + 6 = x^2 - 5x + 6. The other pairings multiply to 6 but give the wrong sum, so they don’t produce the correct middle term when expanded.

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