Problem 63 - Precalculus Mathematics for Calculus

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=x^{5}-x^{4}-5 x^{3}+x^{2}+8 x+4$$

Step-by-Step Solution

Answer

The real zero found via synthetic division is 1. Further factorizations and solving continue from this step.

Step 1: Find Possible Rational Zeros

Use the Rational Root Theorem, which states that any rational zero, or root, of a polynomial is a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term (4) and \( q \) is a factor of the leading coefficient (1). For our polynomial, possible rational zeros are ±1, ±2, ±4.

Step 2: Test Rational Zeros Using Synthetic Division

Out of the possible rational zeros, we need to test each one by performing synthetic division with the polynomial \( P(x) \). We will check if the remainder is zero for each.

Step 3: Identify a Zero

By performing synthetic division, we find that when \( x=1 \), the remainder is zero. Hence, \( x=1 \) is a root of the polynomial.

Step 4: Factor the Polynomial by Dividing

Having identified \( x=1 \) as a root, factor out \( (x-1) \) from \( P(x) \) using synthetic division, which simplifies the polynomial to a lower degree. After this, the polynomial becomes \((x-1)(x^4-5x^3-5x^2+6x+4)\).

Step 5: Repeat Testing of Rational Zeros on Quotient

Repeat Step 2 for the cubic polynomial. After more synthetic division, other rational roots can be identified, if any.

Step 6: Use the Quadratic Formula, If Necessary

After factoring out as much as possible, if you have a quadratic left, for example \( ax^2+bx+c \), then use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find any remaining zeros.

Key Concepts

Polynomial Zeros

The idea of polynomial zeros is central to understanding polynomial equations. A zero of a polynomial is a value for which the polynomial evaluates to zero. In simpler terms, if you substitute a zero back into the polynomial, the result should be zero. These zeros are also known as roots or solutions of the polynomial.

Finding the zeros of a polynomial helps us in factoring the polynomial and sketching its graph. For the polynomial given in the exercise, we need to find all real zeros. We start by using the Rational Root Theorem to list possible zeros, and then proceed to test these candidates to see which ones are actual zeros. This process helps us break down complex polynomials into simpler components.

Synthetic Division

Synthetic division is a simplified form of polynomial division, similar to long division but less tedious. It is specifically useful when dividing a polynomial by a binomial of the form \(x - c\).

To apply synthetic division, follow these steps:

• Write down the coefficients of the polynomial.
• Place the possible zero (from the Rational Root Theorem) to the left.
• Perform the division by bringing down the leading coefficient and then multiplying and adding sequentially with the possible zero.

The primary goal here is to see if the remainder is zero. If it is, the candidate is indeed a zero of the polynomial, and thus \(x-c\) is a factor. This method was used to check for zeros like \(x=1\) in the given polynomial and helped reduce its degree by factoring out \( (x-1) \).

Quadratic Formula

The Quadratic Formula is a powerful tool used to find the zeros of quadratic equations of the form \(ax^2+bx+c=0\). It provides the exact solutions using the equation:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]
This formula is helpful when factoring down polynomials leaves us with a quadratic portion that doesn't factor easily. In the exercise, after employing synthetic division and finding rational zeros, any remaining quadratic can be resolved using this formula.

This formula handles real and complex solutions, making it flexible for various types of expressions. By substituting the coefficients \(a\), \(b\), and \(c\) into the formula, students can compute the roots directly, giving a clear answer to any quadratic leftover from earlier factorizations.