Problem 16 - Precalculus Mathematics for Calculus

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=x^{3}-7 x^{2}+14 x-8$$

Step-by-Step Solution

Answer

The rational zeros are \(1, 2,\) and \(4\); the polynomial in factored form is \((x-1)(x-2)(x-4)\).

Step 1: Identify possible rational zeros

To find the possible rational zeros of the polynomial \( P(x) = x^3 - 7x^2 + 14x - 8 \), we apply the Rational Root Theorem. This theorem states that any rational zero, expressed as a fraction \( \frac{p}{q} \), is such that \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. The constant term here is \(-8\) and the leading coefficient is \(1\). Therefore, the possible rational zeros are the factors of \(-8\), which are \( \pm 1, \pm 2, \pm 4, \pm 8 \).

Step 2: Test the possible rational zeros

We substitute each possible rational zero into \( P(x) \) to determine if it is indeed a zero of the polynomial.- \( P(1) = 1^3 - 7 \cdot 1^2 + 14 \cdot 1 - 8 = 0 \) (So, \( x = 1 \) is a zero.)- \( P(-1) = (-1)^3 - 7 \cdot (-1)^2 + 14 \cdot (-1) - 8 = -30 \)- \( P(2) = 2^3 - 7 \cdot 2^2 + 14 \cdot 2 - 8 = 0 \) (So, \( x = 2 \) is a zero.)- \( P(-2) = (-2)^3 - 7 \cdot (-2)^2 + 14 \cdot (-2) - 8 = -64 \)- \( P(4) = 4^3 - 7 \cdot 4^2 + 14 \cdot 4 - 8 = 0 \) (So, \( x = 4 \) is a zero.)- Since \( x = 8 \) and \( x = -8 \) yield very large non-zero values, these calculations are omitted for brevity.

Step 3: Factor the polynomial using found zeros

We found that \( x = 1, 2, 4 \) are zeros of the polynomial. Therefore, \( P(x) \) can be factored using these zeros: - Use the zero \( x = 1 \) to factor as \( (x - 1) \).- Use the zero \( x = 2 \) to further factor as \( (x - 2) \).- Use the zero \( x = 4 \) to complete the factorization as \( (x - 4) \). Thus, the polynomial in factored form is \( P(x) = (x - 1)(x - 2)(x - 4) \).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a profound yet easy method to narrow down the potential rational zeros (roots) of a polynomial. This theorem proposes that if you have a polynomial equation with integer coefficients, any potential rational root also has to be a fraction that can be formed by dividing a factor of the constant term by a factor of the leading coefficient. In simpler terms, let's say you have a polynomial like \( P(x) = x^3 - 7x^2 + 14x - 8 \). Here, the constant term is \(-8\) and the leading coefficient is \(1\).
Following the theorem:

• The factors of the constant term \(-8\) are \( \pm 1, \pm 2, \pm 4, \pm 8\).
• The factors of the leading coefficient \(1\) are just \( \pm 1\).

Therefore, the possible rational roots must be one of the factors of \(-8\), which are \( \pm 1, \pm 2, \pm 4, \pm 8\). This significantly reduces the number of rational numbers we have to test in finding the zeros.

polynomial factorization

Once you identify the zeros of a polynomial using concepts like the Rational Root Theorem, you can begin the process of factorization. Factorization involves expressing the polynomial as a product of its simpler factors. For example, after finding the zeros for the polynomial \( P(x) = x^3 - 7x^2 + 14x - 8 \), you can write it in its factored form.
Firstly, suppose we found zeros at \( x = 1, 2, 4 \) through substituting and testing possible values. Each of these corresponds to a factor of the polynomial:

• \( x = 1 \) gives \( (x - 1) \).
• \( x = 2 \) gives \( (x - 2) \).
• \( x = 4 \) gives \( (x - 4) \).

So the polynomial, in its factored form, is represented as \( P(x) = (x - 1)(x - 2)(x - 4) \). Factorization is vital because it simplifies complex polynomials and makes them easier to solve and graph.

polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, connected using addition, subtraction, multiplication, and non-negative integer exponents. Take, for instance, \( x^3 - 7x^2 + 14x - 8 \), which is a cubic polynomial because the highest exponent is 3.
Polynomials have several characteristics:

• **Degree:** The highest power (exponent) in a polynomial. This also tells us the maximum number of roots and turning points the polynomial can have.
• **Terms:** Parts of the polynomial separated by plus or minus signs. Each term consists of a coefficient and a variable raised to an exponent.
• **Coefficients:** Numbers that multiply the variables in each term, like \(-7\) in \(-7x^2\).

Understanding these basic attributes of polynomials is key to manipulating them, finding roots, and applying the polynomial for real-world problem-solving. They are foundational elements of algebra, often serving as stepping-stones to more complex mathematical concepts.