Question

Determine whether each binomial is a factor of x^3+3x^2-10x-24

Answer 1

To find if each binomial is a factor of x^3+3x^2-10x-24, one can use the Factor Theorem or perform synthetic or long division with potential factors based on the Rational Root Theorem.

• The student's question revolves around determining whether given binomials are factors of the polynomial x^3+3x^2-10x-24.
• To do this, we can apply the Factor Theorem which states that a binomial (x - c) is a factor of a polynomial if the polynomial evaluates to zero when x is equal to c.
• For instance, if we suspect that (x - 2) is a factor, we substitute x with 2 and evaluate the polynomial.
• If the result is zero, (x - 2) is indeed a factor.
• However, instead of guessing and checking each possible factor, we can use synthetic division or long division to test each possible factor that is a divisor of the constant term, 24, based on the Rational Root Theorem.
• Common factors to check would be ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24 since these are all the divisors of 24.
• Through this process, we can systematically determine which binomials are factors of the given polynomial.
• Perform this analysis for each potential factor until you find all binomials that divide the polynomial without a remainder.

Answer 2

A binomial is a polynomial equation with two terms, each of which is a monomial usually joined by a plus or minus sign. Polynomials with one term is called monomial and could look like this, 9x while a binomial could look like this 2x + 8.

Yes! It true that each binomial is a factor of x^3+3x^2-10x-24. Here’s why,

Quotient: x^2 + 6x + 8
Factor:(x+4)(x+2)

Factor Form: (x-3)(x+4)(x+2)