Problem 50 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Find all real zeros of the polynomial function. \(f(x)=x^5+4 x^4-14 x^3-14 x^2-15 x-18\)

Step-by-Step Solution

Answer

The real zeros of the function \(f(x)=x^5+4 x^4-14 x^3-14 x^2-15 x-18\) are x=1, x=-3, x=-1, and x=-2.

Step 1: Finding Potential Rational Zeros

Start by identifying all the potential rational zeros of the function using the Rational Root Theorem. This theorem states that any potential rational zero \(p/q\) can be found by dividing all factors of the constant term by all factors of the leading coefficient. In this case, both constant and leading coefficients are 1. Therefore, the potential rational zeros are factors of -18, which are ±1, ±2, ±3, ±6, ±9, ±18.

Step 2: Synthetic Division

After identifying the potential zeros, use synthetic division to test them. Start with 1. Setting up the synthetic division table with 1, the coefficients of the polynomial yield a remainder of 0, so 1 is a real zero.

Step 3: Factorization

Now we have a lower-order polynomial to consider, thanks to finding that first zero. The polynomial's degree was reduced by synthetic division. The resulting polynomial is \(x^4+5x^3-9x^2-23x-18\), which can be factored further. Testing rational zeros again (from the Rational Root Theorem) yields x=-3.

Step 4: Find Remaining Zeros

Continuing with the process of synthetic division and factoring, the other real zeros of the function can be found. Continue performing synthetic division for the zeros found until you cannot factor the polynomial any further and all the real zeros are found. By doing so, we find out that the remaining zeros are x=-3, x=-1, and x=-2.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool for finding potential rational zeros of a polynomial. This theorem provides a way to identify these zeros by focusing on integer ratios. Specifically, it states that a polynomial function with integer coefficients may have rational zeros, and these zeros take the form \( \frac{p}{q} \). Here, \( p \) is a factor of the constant term, while \( q \) is a factor of the leading coefficient.

• Identify the potential numerators, \( p \), by listing all factors of the constant term.
• Identify the potential denominators, \( q \), by listing all factors of the leading coefficient.
• Generate all possible fractions \( \frac{p}{q} \) to find potential rational zeros.

In our exercise, since both the constant and leading coefficients were 1, the potential rational zeros were simply the factors of -18. These potential zeros included ±1, ±2, ±3, ±6, ±9, and ±18.

Synthetic Division

Synthetic Division is a simplified method of polynomial division, particularly when dealing with linear divisors of the form \(x - c\). This process is more streamlined compared to long division and is primarily used to test potential zeros or perform factorization.

• Create a row with the coefficients of the polynomial.
• Use the potential zero to set up the division.
• Multiply, add, and bring down the coefficients iteratively to compute the result.

If the remainder is zero, the divisor is a root of the polynomial. For example, in the exercise above, we tested potential zeros using synthetic division. When testing 1 and finding a remainder of zero, we confirmed that 1 is a real zero. This allows us to reduce the polynomial’s degree and continue searching for other zeros.

Real Zeros

Real zeros of a polynomial function are the inputs (x-values) that make the entire function equal to zero. In simpler terms, these are the points where the graph of the polynomial crosses or touches the x-axis.Finding real zeros typically involves testing all potential rational zeros and factoring. In the exercise, we started with a fifth-degree polynomial, specifically \(f(x)=x^5+4x^4-14x^3-14x^2-15x-18\). Through methods like synthetic division and rational root testing, we reduced this polynomial step by step.Ultimately, we found the real zeros to be x = 1, x = -3, x = -1, and x = -2. These zeros can tell us a lot about the polynomial, such as its behavior and the number of times the graph intersects the x-axis.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are written in the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where \( n \) is a non-negative integer, and \( a_n \) is not equal to zero.

• Each term consists of a coefficient and a variable raised to an exponent.
• The highest exponent indicates the degree of the polynomial.
• Polynomial functions can have complex, real, and rational zeros.

In our example, the polynomial \(f(x)=x^5+4x^4-14x^3-14x^2-15x-18\) is of degree 5, which means it can have up to 5 real solutions, including both distinct and repeated roots. Understanding the characteristics of polynomials allows us to solve equations, model real-world scenarios, and analyze functions effectively.