Problem 11 - Algebra 2

Question

Determine whether each binomial is a factor of \(x^{3}+4 x^{2}+x-6\) $$ x+3 $$

Step-by-Step Solution

Answer

Using the Factor Theorem, after substituting -3 into the polynomial and finding that it equals zero, we confirm that x + 3 is a factor of the polynomial.

Step 1: Understand the Factor Theorem

The Factor Theorem states that if a polynomial f(x) is divisible by a binomial (x - c), then c is a root of the polynomial. Similarly, if x + c is a factor, then -c is a root. For the binomial x + 3 to be a factor of the polynomial, -3 must be a root of the polynomial.

Step 2: Apply the Factor Theorem

Substitute the value -3 into the polynomial in place of x and evaluate: f(-3) = (-3)^{3} + 4(-3)^{2} + (-3) - 6.

Step 3: Calculate the Value of f(-3)

f(-3) = (-27) + 4(9) - 3 - 6 = -27 + 36 - 3 - 6 = 0. Since f(-3) = 0, the binomial x + 3 is indeed a factor of the polynomial.

Key Concepts

Polynomial Factorization

The process of breaking down a polynomial into simpler components that, when multiplied together, give back the original polynomial is known as polynomial factorization. It's like finding a recipe for a cake; each ingredient represents a factor, and when combined, they recreate the entire cake, which in this case, is the polynomial equation.

Polynomial factorization is useful because it simplifies complex polynomials into products of binomial or smaller polynomial factors, which makes dealing with equations more manageable especially when solving for zeros, graphing polynomials, or integrating. The Factor Theorem plays a crucial role in polynomial factorization as it assists in finding these factors. When a polynomial is completely factored, it may look like a series of binomials multiplied together, each presenting a root of the polynomial, which leads to most effective strategies in problem-solving.

Binomial Factors

A binomial factor in the context of polynomials is a two-term expression such as (x - a) or (x + b), where 'a' and 'b' are constants. These binomial factors are significant since they reveal the roots of the polynomial. This connection is underscored by the Factor Theorem, which provides a direct method to test whether a given binomial is a factor of a polynomial.

When a binomial such as (x + b) is a factor of a polynomial, it signifies that the polynomial has a root at x = -b. This is because setting the binomial equal to zero and solving for x will yield this particular value. Testing for binomial factors can be performed by substituting the potential root into the polynomial to see if it evaluates to zero, confirming that the binomial is indeed a factor.

Roots of Polynomials

The roots of a polynomial are the values for which the polynomial equation equals zero. These are the exact points where the graph of the polynomial intersects the x-axis. Finding the roots is a primary concern in algebra because it allows us to solve equations and understand the behavior of polynomial functions.

The connection between roots and binomial factors is made clear by the Factor Theorem, which states that if a number 'c' is a root of the polynomial f(x), the binomial (x - c) is a factor of f(x). Conversely, if a binomial (x - c) is a factor of f(x), then 'c' is a root. Root finding can sometimes be approached by graphing, through various numerical methods, or by polynomial factorization which simplifies the problem into a set of binomial factors that can be easily solved.