Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Find all the zeros of the function and write the polynomial as a product of linear factors. $$g(x)=x^{5}-8 x^{4}+28 x^{3}-56 x^{2}+64 x-32$$

Step-by-Step Solution

Answer

The root of the function is \( x = 2 \) with multiplicity 5. Hence the factored form is \( g(x) = 32(x-2)^5 \).

Step 1: Identify the Binomial Expression

Notice the coefficients in the polynomial. They form a pattern that repeats 1, 4, 6, 4, 1. This is a pattern associated with the binomial theorem. Accordingly, the original polynomial can be formed by expanding binomial expression \( (x-2)^5 \).

Step 2: Expand Binomial Expression

Expand the binomial expression, \( (x-2)^5 \), which equals to \( x^{5}-5x^{4}+10x^{3}-10x^{2}+5x-1 \). If we multiply this expression by 32, we will get the original polynomial.

Step 3: Identify Zeros and Write Polynomial in its Factored Form

The zero of the polynomial is the value of x for which \( g(x) = 0 \). In this case, \( (x-2)^5 = 0 \), so \( x = 2 \) is the only root of the function, occurring five times. Hence the factored form of the given polynomial is \( g(x) = 32(x-2)^5 \).

Key Concepts

Zeros of a Function

Zeros of a function are critical points where the function value equals zero. In simpler terms, they are the solutions to the equation \( f(x) = 0 \). These zeros are important because they tell us where the graph of the function will intersect the x-axis. For our polynomial \( g(x) = x^{5}-8x^{4}+28x^{3}-56x^{2}+64x-32 \), finding the zeros is the first step in solving the problem.

In the given exercise, we determined that \( x = 2 \) is a zero of the function. This means when \( x \) is substituted with 2 in the polynomial, the result is zero. Furthermore, since \( (x-2)^5 \) shows that this root repeats five times, we call it a root with multiplicity 5. This concept highlights that a zero can occur more than once, affecting the shape of the polynomial's graph.

Binomial Expansion

The binomial expansion is a technique that allows us to expand expressions that are powers of a binomial like \( (x - a)^n \). This method uses the binomial theorem which involves coefficients that follow a specific pattern known as Pascal's triangle.

In our problem, the polynomial can be created by expanding \( (x-2)^5 \), which results in \( x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 \). They follow the coefficients 1, 5, 10, 10, 5, 1, matching the rows of Pascal's triangle starting from row zero. Notice that these patterns help in identifying potential factors of a polynomial, transforming a complex expression into a more manageable one.

Factored Form

The factored form of a polynomial expresses the function as a product of its factors. Factors can be binomials or other algebraic expressions that, when multiplied together, will yield the original polynomial. Factoring is a key skill for simplifying expressions and solving polynomial equations.

For \( g(x) = x^{5}-8x^{4}+28x^{3}-56x^{2}+64x-32 \), we expressed it as \( g(x) = 32(x-2)^5 \) in its factored form. This indicates that 32 is a scaling factor, and \( (x-2) \) is a repeated factor because it existed five times, correlating with the root's multiplicity. Factoring the polynomial helps us see the zeros and their multiplicities clearly, and dramatically simplifies solving and graphing the polynomial function.