(x+2)(x-3)(x+5) Is Identical To X^3+ax^2-11x+b

Jasonbradley

2 min read

·

Jun 16, 2024

46

Share

Expanding and Comparing Polynomials: (x+2)(x-3)(x+5) and x^3 + ax^2 - 11x + b

This article explores the relationship between the expanded form of the polynomial (x+2)(x-3)(x+5) and the polynomial x^3 + ax^2 - 11x + b. Our goal is to determine the values of a and b that make the two expressions identical.

Expanding the Polynomial

First, we expand the product (x+2)(x-3)(x+5) using the distributive property (or FOIL method):

1. Expand (x+2)(x-3): (x+2)(x-3) = x² - x - 6

2. Multiply the result by (x+5): (x² - x - 6)(x+5) = x³ + 4x² - 11x - 30

Therefore, the expanded form of (x+2)(x-3)(x+5) is x³ + 4x² - 11x - 30.

Comparing Coefficients

Now we compare the expanded form with x³ + ax² - 11x + b:

• x³ coefficient: Both expressions have a coefficient of 1 for the x³ term.
• x² coefficient: The expanded form has a coefficient of 4 for the x² term, while the other expression has a coefficient of a. Therefore, a = 4.
• x coefficient: Both expressions have a coefficient of -11 for the x term.
• Constant term: The expanded form has a constant term of -30, while the other expression has a constant term of b. Therefore, b = -30.

Conclusion

We conclude that the polynomial (x+2)(x-3)(x+5) is identical to x³ + 4x² - 11x - 30. Therefore, a = 4 and b = -30.