Question

What are the possible integral zeros of each polynomial? a) \(P(x)=x^{3}+3 x^{2}-6 x-8\) b) \(P(s)=s^{3}+4 s^{2}-15 s-18\) c) \(P(n)=n^{3}-3 n^{2}-10 n+24\) d) \(P(p)=p^{4}-2 p^{3}-8 p^{2}+3 p-4\) \(P(z)=z^{4}+5 z^{3}+2 z^{2}+7 z-15\) f) \(P(y)=y^{4}-5 y^{3}-7 y^{2}+21 y+4\)

Short answer

For polynomial a): -8, -1, 1, 2, 4, 8, polynomial b): -18, -9, -6, -3, -2, 2, 3, 6, 9, 18, polynomial c): -24, -12, -8, -6, -4, -3, -2, 1, 2, 3, 4, 6, 8, 12, 24, polynomial d): -4, -2, -1, 1, 2, 4, polynomial e): -15, -5, -3, -1, 1, 2, 3, 5, 15, polynomial f): -4, -2, -1, 1, 2, 4.

Step-by-step solution

Identify Constant and Leading Coefficients

For each polynomial, identify the constant term (the term without the variable) and the leading coefficient (the coefficient of the highest-degree term).

Apply Rational Root Theorem

The Rational Root Theorem states that any possible rational root, in the form of a fraction \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. Here, focus only on integral (whole number) roots.

List Factors of Constant Term

List all factors (positive and negative) of the constant term. These will be checked as possible integral zeros.

Substitute Possible Zeros into Polynomial

Substitute each of the possible integral zeros into the polynomial equation and solve to see if it equals zero. If the polynomial equals zero, then that factor is a root.

Repeat for Each Polynomial

Repeat the above steps for each polynomial given in the problem.

Key concepts

Rational Root Theorem

The Rational Root Theorem is a key concept in finding potential rational zeros for polynomials. A rational root is a root that can be expressed as a fraction. This theorem states that if a polynomial has a rational root expressed as \( \frac{p}{q} \), then the numerator \( p \) is a factor of the constant term, and the denominator \( q \) is a factor of the leading coefficient. Since we are looking for integral zeros, the potential roots are simply the factors of the constant term divided by the factors of the leading coefficient listed as whole numbers.
To apply it:

• Identify the constant term in the polynomial.
• Identify the leading coefficient of the polynomial.
• List all integer factors of the constant term.
• Check these factors by substituting them into the polynomial to verify if they satisfy the equation.

This theorem helps narrow down the list of potential zeros, making the problem easier to solve.

Polynomial Roots

Polynomial roots are values for the variable that make the polynomial equal to zero. These roots can be real or complex, and in many problems, we specifically focus on finding real roots.
For instance, if given a polynomial like \( P(x) = x^3 + 3x^2 - 6x - 8 \), finding a root means solving \( P(x) = 0 \).
Here's how to start:

• Use the list of possible integral zeros.
• Substitute each candidate into the polynomial.
• If the result is zero, that candidate is a root.
• Factoring the polynomial using the found roots can help identify other possible roots.

Understanding polynomial roots is crucial for solving equations, simplifying expressions, and graphing functions.

Mathematical Substitution

Substitution is a fundamental technique in algebra used to find the value of variables. When testing potential zeros for a polynomial, we substitute these values into the polynomial equation to see if it equals zero.
For example, if we have a polynomial \( P(x) = x^3 + 3x^2 - 6x - 8 \), and we want to test if \( x = -2 \) is a zero, we substitute \( -2 \) for \( x \):
\[ P(-2) = (-2)^3 + 3(-2)^2 - 6(-2) - 8 = -8 + 12 + 12 - 8 = 8 eq 0 \]
Since \( P(-2) eq 0 \), \( -2 \) is not a root. The method of substitution allows us to confirm whether a value is a zero of the polynomial in a straightforward manner.
This helps us systematically test each potential root, one by one.

Factors of Polynomials

A polynomial's factors are expressions that, when multiplied together, give the original polynomial. Finding the polynomial's roots helps in determining its factors. For example, if \( x = 2 \) is a root of the polynomial \( P(x) \), then \( x - 2 \) is a factor.
Here's a step-by-step process:

• Identify a root of the polynomial using substitution or the Rational Root Theorem.
• Use polynomial division to divide the original polynomial by \( x - \text{root} \).
• Repeat the process with the resulting polynomial until it's completely factored.

Factoring a polynomial can simplify the equation and solve for its zeros more efficiently.
In summary, the factors of a polynomial provide an insightful way to break down and understand the structure of polynomial equations, which is essential for deeper mathematical analysis and simplifying problems.