The second step in the process for factoring the trinomial x^2-3x-40 is to
The Correct Answer and Explanation is:
Correct Answer:
The second step in factoring the trinomial x2−3x−40x^2 – 3x – 40×2−3x−40 is to find two numbers that multiply to give -40 and add to give -3.
Explanation:
Factoring a trinomial such as x2−3x−40x^2 – 3x – 40×2−3x−40 involves expressing it as a product of two binomials. This process typically starts by identifying the structure of the expression. Since the leading coefficient is 1, the goal becomes identifying two integers that satisfy specific conditions: their product must be equal to the constant term, which is -40, and their sum must match the coefficient of the middle term, which is -3.
To proceed, consider the factors of -40. Since the product is negative, one number must be positive and the other negative. Begin listing factor pairs of -40:
- (1, -40), (-1, 40)
- (2, -20), (-2, 20)
- (4, -10), (-4, 10)
- (5, -8), (-5, 8)
Among these pairs, the set that adds up to -3 is (5, -8). This satisfies both requirements:
- 5×(−8)=−405 \times (-8) = -405×(−8)=−40
- 5+(−8)=−35 + (-8) = -35+(−8)=−3
Identifying this correct pair of numbers is the second step because it provides the foundation for rewriting the middle term, which sets up the factoring by grouping in the next stage. This technique is efficient and particularly useful when the coefficient of the squared term is 1.
Mastering this step helps in solving quadratic equations, simplifying algebraic expressions, and understanding how expressions can be reversed from expanded form into factored form. Recognizing number pairs that meet both conditions requires practice and familiarity with multiplication and addition patterns.
