Question

Find all the zeros of the function and write the polynomial as a product of linear factors. $$h(x)=x^{3}+9 x^{2}+27 x+35$$

Short answer

The roots of the function \(h(x)=x^{3}+9 x^{2}+27 x+35\) are -1,-7 and -5. Polynomial as a product of linear factors is \(h(x)=(x+1)(x+7)(x+5)\).

Step-by-step solution

Use the Rational Root Theorem

The Rational Root Theorem states that any rational root, expressed in the form \(p/q\), where p and q are integers and q ≠ 0, of the polynomial equation \(ax^n + bx^{n-1} + ... k = 0\) is such that p is a factor of k (the constant term) and q is a factor of a (the leading coefficient). Here, a=1, k=35.

Guess & Check for Potential Rational Zeroes

List all the factors of a and k and form potential rational roots. For a=1, factors={1}; for k=35, factors={±1, ±5, ±7, ±35}. This gives potential roots as {±1, ±5, ±7, ±35}.

Substitute Potential Roots into the Equation

Substitute potential roots into the equation and check which one makes the equation equal to zero. For \(h(x)=x^{3}+9x^{2}+27x+35\), when x=-1, \(h(x)=(-1)^{3}+9(-1)^{2}+27(-1)+35=0\). So, x=-1 is a root.

Conduct Polynomial Division

Perform polynomial division or use synthetic division to divide the polynomial by the factor associated with the discovered root, x = -1. The result will be a quadratic function.

Find the Remaining Roots

Find the roots of the resulting quadratic equation by using the quadratic formula, \(x=\frac{-b±\sqrt{b^2-4ac}}{2a}\). The roots from the quadratic equations are the remaining roots of the original function.

Write the Polynomial as Linear Factors

The original function can be re-written as the product of linear factors each represented by x-root.

Key concepts

Rational Root Theorem

The Rational Root Theorem is an invaluable tool when you are faced with a polynomial like h(x) = x^3 + 9x^2 + 27x + 35 and need to find its zeros. This theorem provides a way to list all possible rational zeros of a polynomial equation. These potential zeros are of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In cases where the leading coefficient is 1, the possible zeros are simply the factors of the constant term.

For the polynomial given, since the leading coefficient is 1, the Rational Root Theorem tells us that any rational zero must be a factor of 35. Thus, you check each factor to see if it is indeed a zero of the polynomial. This can narrow down the search significantly, saving time and effort.

Finding these candidates is just the first step—verifying which ones are actual zeros requires substituting them into the polynomial and checking for a zero result, as was done for the factor -1.

Polynomial Division

Once a zero is confirmed using the Rational Root Theorem, the next step is to simplify the polynomial, and that's where Polynomial Division comes into play. By dividing the polynomial by (x - zero), you effectively reduce its degree by one, getting one step closer to breaking it down to all of its linear factors. In our exercise, after finding that -1 is a zero, we divide the polynomial h(x) by (x + 1)..

Polynomial Division can be done through long division or synthetic division, both methods yielding the same result. Completing the division gives us a smaller polynomial to work with, which is easier to handle, especially when looking for additional zeros. In this case, completing the division would result in a quadratic equation, which we can then solve using other methods.

Quadratic Formula

When Polynomial Division reduces the original polynomial to a quadratic, we can use the Quadratic Formula to find the remaining zeros. This formula, x = ( -b ± √(b^2 - 4ac) ) / 2a, comes to the rescue when we're dealing with quadratic equations that may not factor easily. It is a universal method that works for any quadratic equation.

The formula provides the solutions by considering the coefficients of the quadratic equation derived from the polynomial. These coefficients come from the standardized form of a quadratic equation, ax^2 + bx + c = 0. This dependable method ensures that you don't miss any real zeros and also helps identify if there are complex roots, which is incredibly useful in more advanced mathematics or when the quadratic doesn't factor nicely.

Linear Factors

Turning a polynomial into a product of Linear Factors boils down to expressing the polynomial as (x - r1)(x - r2)...(x - rn), where each r represents a zero of the polynomial. It's a neat way of displaying the roots, whether they're rational, irrational, or even complex, in a clean and straightforward manner.

In our initial function h(x), once all the zeros are found using methods like the Rational Root Theorem and the Quadratic Formula, we can write out the polynomial in terms of its linear factors. Each zero found gives a factor of (x - zero), and when multiplied together, they recreate the original polynomial. This representation not only reveals all the roots at a glance but also confirms our solutions in an elegant and satisfying product, giving closure to the fact that all the zeros have been accounted for.