Problem 26 - Precalculus: Functions and Graphs

Question

Show that the number is a zero of \(f(x)\) of the given multiplicity, and express \(f(x)\) as a product of linear factors. $$\begin{aligned} f(x)=x^{5}+x^{4}-6 x^{3}-14 x^{2}-11 x-& 3 \\ -1 &-1 \text { (multiplicity 4) } \end{aligned}$$

Step-by-Step Solution

Answer

The polynomial is factored as \((x+1)^4(x-3)\).

Step 1: Verify the Root

First, verify that -1 is a root of multiplicity 4. This means when you plug -1 into the function, the result should be zero, and -1 should satisfy the polynomial equation a total of 4 times.

Step 2: Perform Polynomial Division

Perform synthetic or long division of the polynomial \( f(x) = x^5 + x^4 - 6x^3 - 14x^2 - 11x - 3 \) by \( x + 1 \) multiple times. This is repeated 4 times because the root -1 is given with a multiplicity of 4. This process will reduce the polynomial by 4 degrees.

Step 3: Obtain Quotient Polynomial

After performing the division 4 times, observe the quotient polynomial remaining from the division. This polynomial should be of degree 1.

Step 4: Express as a Product of Linear Factors

Express the original polynomial \( f(x) \) as a product of the linear factors obtained, which includes \((x + 1)^4\) and the quotient polynomial from the division.

Step 5: Verify Factorization

Multiply the linear factors to ensure that you obtain the original polynomial, verifying the factorization is correct.

Key Concepts

Roots of Polynomials

Polynomials are expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. A 'root' of a polynomial is a solution to the equation when the polynomial is set equal to zero. In simpler terms, a root is a value that when substituted into the polynomial, makes the entire expression zero.
- For example, if you have a polynomial like \( f(x) = x^5 + x^4 - 6x^3 - 14x^2 - 11x - 3 \), then finding a root of this polynomial involves determining the value of \( x \) where \( f(x) = 0 \).- Roots can be real or complex numbers. Real roots are those values that you can plot on a number line, while complex roots include imaginary numbers.
In the given exercise, the number -1 is verified as a root by plugging it into the polynomial \( f(x) \). If the output is zero, which it is, then -1 is indeed a root of the polynomial.

Multiplicity of Roots

Multiplicity refers to the number of times a particular root appears as a solution to the polynomial equation. If a root has a multiplicity greater than 1, it means the graph of the polynomial touches the x-axis at this root, but doesn’t cross it.
- A root with a multiplicity of 1 crosses the x-axis at that point.- A root with a multiplicity of 2 appears as a "bounce" on the x-axis.
In the exercise, it is stated that the root -1 has a multiplicity of 4. This means the original polynomial can be divided by \( (x + 1) \) a total of 4 times. This multiplicity indicates that -1 is not only a root but plays a significant role in the structure of the polynomial, affecting its graph by making it touch, but not cross, at x = -1.

Synthetic Division

Synthetic division is a simplified way for dividing a polynomial by a binomial of the form \( x - c \). It allows for quick calculations by using only the coefficients of the polynomial. This method is especially efficient when repeatedly dividing by the same binomial.
To perform synthetic division for our example, follow these steps:

• Write down all the coefficients of \( f(x) \).
• Since we are dividing by \( x + 1 \), use \( -1 \) in the synthetic division setup.
• Bring down the leading coefficient; then, multiply it by \( -1 \) and add to the next coefficient. Repeat for the remaining coefficients.

This method must be repeated 4 times for this example, as -1 is a root with a multiplicity of 4. After four steps, you will achieve a new polynomial quotient reduced so that the entire degree of the original polynomial has been reduced by 4.

Linear Factors

Linear factors are expressions of the form \( x - r \), where \( r \) is a root of the polynomial. Factoring a polynomial into its linear factors completely breaks it down into these components.
When you express a polynomial as a product of its linear factors, you effectively represent all its roots. Ultimately, the goal of factorization is:

• To break down the polynomial into simpler expressions.
• To provide clear insights into the roots and their multiplicities.

In the current exercise, after repeatedly performing synthetic division, the polynomial \( f(x) \) can be expressed as a product of linear factors: - Starting with \((x + 1)^4 \) to represent the repeated root -1.- Include any remaining polynomial factor resulting from the division process.
This makes it easy to write \( f(x) \) as a product: \( f(x) = (x + 1)^4 \times (remaining ext{ } factor) \), which simplifies understanding and solving polynomial functions.