Factor each polynomial completely. See Example 6. x² - 2x - 15

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Identify the quadratic polynomial to factor: \(x^{2} - 2x - 15\).

Recall that to factor a quadratic of the form \(x^{2} + bx + c\), we look for two numbers that multiply to \(c\) and add to \(b\).

Find two numbers that multiply to \(-15\) and add to \(-2\). Consider the factor pairs of \(-15\): \((1, -15)\), \((-1, 15)\), \((3, -5)\), and \((-3, 5)\).

Determine which pair sums to \(-2\). The pair \((3, -5)\) multiplies to \(-15\) and adds to \(-2\).

Write the factored form using these numbers: \((x + 3)(x - 5)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Factoring Quadratic Polynomials

Factoring quadratic polynomials involves expressing a quadratic expression as a product of two binomials. For a quadratic in the form x² + bx + c, the goal is to find two numbers that multiply to c and add to b. This process simplifies solving equations and analyzing functions.

Finding Factors of the Constant Term

To factor a quadratic, identify pairs of factors of the constant term (here, -15) that combine to give the middle coefficient (-2). For example, factors of -15 include (1, -15), (-1, 15), (3, -5), and (-3, 5). Selecting the correct pair is essential for accurate factoring.

Zero Product Property

Once factored, the quadratic can be set equal to zero and solved using the zero product property, which states that if a product of factors equals zero, at least one factor must be zero. This property is fundamental for solving quadratic equations after factoring.