In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x²+8x+15

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Identify the trinomial in the form \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = 8 \), and \( c = 15 \).

Look for two numbers that multiply to \( c = 15 \) and add to \( b = 8 \). These two numbers will be used to split the middle term.

The two numbers that satisfy the conditions are 5 and 3, because \( 5 \times 3 = 15 \) and \( 5 + 3 = 8 \).

Rewrite the trinomial by splitting the middle term using the two numbers found: \( x^2 + 5x + 3x + 15 \).

Group the terms into two pairs and factor each group: \( (x^2 + 5x) + (3x + 15) \), then factor out the greatest common factor (GCF) from each group.

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Factoring Trinomials

Factoring trinomials involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to 'c' (the constant term) and add to 'b' (the coefficient of the linear term). For example, in the trinomial x^2 + 8x + 15, we look for two numbers that multiply to 15 and add to 8.

Prime Trinomials

A trinomial is considered prime if it cannot be factored into the product of two binomials with real coefficients. This typically occurs when the discriminant of the quadratic equation is negative or when no integer pairs satisfy the conditions for factoring. Recognizing prime trinomials is essential for determining whether a quadratic expression can be simplified further.

Quadratic Formula

The quadratic formula, given by x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of a quadratic equation ax^2 + bx + c = 0. While not directly used for factoring, it helps determine if a trinomial can be factored by revealing the nature of its roots. If the roots are rational, the trinomial can be factored; if they are irrational or complex, it may be prime.

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x2+8x+15