Problem 55 - Precalculus Mathematics for Calculus

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=2 x^{4}+15 x^{3}+17 x^{2}+3 x-1$$

Step-by-Step Solution

Answer

The real zeros are estimated using graphing/numerical tools.

Step 1: Identify the Polynomial and Its Degree

The given polynomial is \(P(x) = 2x^4 + 15x^3 + 17x^2 + 3x - 1\). This polynomial is of degree 4, which means it can have up to 4 real zeros.

Step 2: Check for Rational Roots Using Rational Root Theorem

Using the Rational Root Theorem, we list potential rational roots, which are \(\pm 1\). We substitute these values into the polynomial:\[P(1) = 2(1)^4 + 15(1)^3 + 17(1)^2 + 3(1) - 1 = 36\]\[P(-1) = 2(-1)^4 + 15(-1)^3 + 17(-1)^2 + 3(-1) - 1 = -18\]Neither of these are zero, so there are no rational roots.

Step 3: Apply Synthetic Division to Simplify the Polynomial

Since the rational root check failed, we attempt to simplify and factor the polynomial. First, try synthetic division to guess and check possible factors. Start with \(x = -0.5\). Use synthetic division to check:After performing the synthetic division with \(-0.5\), we find it doesn't simplify neatly, suggesting it's likely not an exact root.

Step 4: Use Approximations or Graphing Tools

Since factoring analytically is not feasible after my synthetic division attempts, using a graphing tool such as a graphing calculator or software can help estimate real zeros. Graphically, it helps to approximate or check using additional methods like Newton's method if needed.

Step 5: Using the Quadratic Formula

If the polynomial can be split into quadratic segments (if using synthetic division gave usable quotients), use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) on any simpler quadratic segments. However, based on no obvious factorization, we'll rely more on approximations for numerical values beyond our manual checks. Typically, the zeros are approximated or found using numerical computational tools.

Key Concepts

Degree of Polynomial

The degree of a polynomial is one of its most important characteristics. It represents the highest power of the variable in the equation. In the given polynomial \[P(x) = 2x^4 + 15x^3 + 17x^2 + 3x - 1,\] the degree is 4, as indicated by the highest exponent of the term \(x^4\). The degree tells us a lot about the polynomial:

• A polynomial of degree 4 can have up to 4 real zeros, or roots.
• The end behavior of the graph is determined by the leading term, such as \(2x^4\), indicating the direction the graph extends towards infinity.

Understanding the degree helps us set expectations for the complexity involved in finding the roots of the polynomial.

Rational Root Theorem

The Rational Root Theorem is a handy tool for estimating possible rational roots of a polynomial with integer coefficients. For our polynomial \[P(x) = 2x^4 + 15x^3 + 17x^2 + 3x - 1,\] potential rational roots can be determined by the factors of the constant term (\(-1\)) and the factors of the leading coefficient (\(2\)). The theorem gives us the possible rational roots

• The possible roots are in the form \(\pm\frac{1}{1}, \pm\frac{1}{2}\).

However, upon substituting these values back into the polynomial, we discover none of them yields zero, suggesting there are no rational roots. This insight is crucial as it guides us towards potentially more complex strategies like synthetic division or numeric methods.

Synthetic Division

Synthetic division acts as a simplified form of traditional long division for determining whether a certain value is a root of a polynomial. It is particularly useful for simplifying calculations and reducing polynomials when testing possible roots derived from the Rational Root Theorem. To perform synthetic division on \[P(x) = 2x^4 + 15x^3 + 17x^2 + 3x - 1,\] with a potential root of \(-0.5\), follow these steps:

• Use coefficients \(2, 15, 17, 3, -1\).
• When \(-0.5\) is used as a divisor, issues in simplification suggest it isn't a clear root.

While synthetically dividing, if no exact integer or rational solutions emerge, the polynomial needs either factorization into quadratics or graphical/numerical methods for root evaluation.

Quadratic Formula

The quadratic formula comes into play when a polynomial can be reduced to a quadratic form after simplification. If our polynomial \[P(x) = 2x^4 + 15x^3 + 17x^2 + 3x - 1,\] is successfully factored or reduced into quadratic components, the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] is invaluable for finding the precise zeros analytically. However, this particular polynomial does not easily factor into quadratic segments, so while the quadratic formula represents a powerful approach for simpler quadratic pieces, in this case, numerical methods or graphing techniques are often preferred to estimate zeros. The inability to use the formula here highlights the complexities sometimes faced when handling higher-degree polynomials.