Problem 37 - College Algebra

Question

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=x^{3}-x^{2}+25 x-25 $$

Step-by-Step Solution

Answer

The zeros of the polynomial are \(x = 1\) (multiplicity of 2) and \(x = -5\) (multiplicity of 1). The polynomial can be expressed as a product of linear factors: \(f(x) = (x - 1)^2(x + 5)\)

Step 1: Find the zeroes

Start by setting the polynomial equal to zero and solve the resulting equation for x. Doing so gives us: \(x^{3}-x^{2}+25x-25 = 0\)

Step 2: Factor by Grouping

The next step is to apply the method of factor by grouping, but this polynomial cannot be grouped as such. So, the next option is to try to factor by synthetic division or rational root theorem.

Step 3: Apply Rational Root Theorem

The Rational Root theorem states that if a polynomial has a rational root \(p/q\), then \(p\) is a factor of the trailing constant and \(q\) is a factor of the leading coefficient. For this polynomial the options for \(p/q\) are \(\pm 1, \pm 5, \pm 25\). Checking these options, we find that \(1\) and \(-5\) are zeroes of the polynomial.

Step 4: Perform Synthetic Division

Now, let's do synthetic division for the above-found roots. We divide our polynomial \(x^3 - x^2 + 25x - 25\) by the binomial \((x - 1)\) to get a new polynomial. We then divide this new polynomial by \((x + 5)\), so our polynomial simplifies down to \(x - 1\). Now, our polynomial is \(f(x) = (x - 1)(x + 5)(x - 1)\).

Step 5: Write Polynomial as Product of Linear Factors

Now that we have our roots and the simplified polynomial, we can rewrite the original polynomial as a product of linear factors. The polynomial as a product of linear factors is: \(f(x) = (x - 1)(x + 5)(x - 1)\)

Key Concepts

synthetic division

Synthetic division is a method used to divide a polynomial by a binomial of the form \((x-c)\). It's a simplified version of long division, which is especially useful when dealing with polynomials. Synthetic division is much faster and requires fewer steps, which makes it accessible for solving polynomial roots.

To perform synthetic division, follow these steps:

• Write down the root of the divisor in the form \(x-c\). For our problem, the roots were \(1\) and \(-5\).
• List the coefficients of the polynomial to be divided. For example, with \(f(x) = x^3 - x^2 + 25x - 25\), the coefficients are \(1, -1, 25, -25\).
• Drop the leading coefficient down as is.
• Multiply this leading coefficient by the root of the divisor \(c\) and add this product to the next coefficient.
• Continue this process through the list of coefficients, replacing each with the result.
• The final number will be the remainder, which should be zero if \(c\) truly is a root.

Synthetic division not only helps confirm roots but also simplifies the polynomial each time a division is performed.

rational root theorem

The Rational Root Theorem is a valuable tool for finding possible rational roots of a polynomial. It provides a way to test whether potential rational numbers are solution candidates. This theorem states that any rational root, expressed as \(p/q\), must satisfy the condition that \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.

In our example, the polynomial \(x^3 - x^2 + 25x - 25\) has potential rational roots from the factors of the constant term \(-25\) and the leading coefficient \(1\). The possible rational roots can be:

• \(\pm 1\)
• \(\pm 5\)
• \(\pm 25\)

By substituting each potential root back into the polynomial, we identified that \(1\) and \(-5\) are indeed roots. This technique is efficient for narrowing down possible candidates, allowing further exploration such as synthetic division to confirm the roots.

factoring polynomials

Factoring polynomials means rewriting them as a product of simpler polynomials. This helps in solving polynomial equations by breaking them down into factors that are either linear or easily solvable. Once the polynomial is expressed in its factored form, each factor can be set to zero to find the roots.

The polynomial \(f(x) = x^3 - x^2 + 25x - 25\) can be factored using the synthetic division results and the rational roots found earlier.
First, confirm the factors using synthetic division:

• Divide the polynomial by \((x-1)\) using synthetic division.
• With the quotient, perform another synthetic division using \((x+5)\).

The quotient will be a simple linear polynomial or constant. With all roots verified, express the polynomial as a product of linear factors: \(f(x) = (x-1)(x+5)(x-1)\).
Factoring offers an effective way to find all roots, simplify expressions, and solve equations.