Problem 35 - Precalculus : Building Concepts and Connections

Question

Determine together \(q(x)\) is a factor of \(p(x)\) Here, \(p(x)\) is the first polynomial and \(q(x)\) is the second polynomial. justify your answer. $$x^{4}-50 ; x-5$$

Step-by-Step Solution

Answer

After performing synthetic division using the root of \(q(x)\) on \(p(x)\), we find that the remainder is 0, therefore, \(x-5\) is indeed a factor of \(x^{4} - 50.\)

Step 1: Determine the coefficients of the polynomials

For the polynomial \(p(x) = x^{4} - 50\), the coefficients in descending order of powers of \(x\) are - [1, 0, 0, 0, -50]. For the polynomial \(q(x) = x-5\), because the term is linear, the root is equal to 5 (we can find this by setting the polynomial equal to zero).

Step 2: Execution of synthetic division

Synthetic division is executed as follows: First, write the root (5) in a box or a small circle to the left. Then, write the coefficients of the given polynomial to the right, underneath which we will perform the synthetic division operations. After setting the first coefficient (1) directly down, multiply the root (5) by the copied coefficient (1), and put the product under the second coefficient (0) and add to get the new line of numbers. Repeat the multiplication and addition process until all coefficients are covered. We should get a 0 at the end, because we're looking at whether or not 5 is a root.

Step 3: Verify the remainder

Verify the remainder of the synthetic division operation. If the remainder is 0, \(q(x)\) is a factor of \(p(x)\) because by the Factor Theorem, a polynomial \(f(x)\) has a factor \(x-r\) if and only if \(f(r)=0\).

Key Concepts

Factor Theorem

The Factor Theorem is an essential tool when working with polynomials. This theorem tells us that a polynomial \(p(x)\) has a factor \((x - r)\) if and only if \(p(r) = 0\). This means that if you can plug a number "r" into the polynomial and it equals zero, \(x - r\) is a factor of that polynomial.

This concept is very handy when you need to determine the roots of a polynomial or check if a particular linear expression is a factor. Essentially, the Factor Theorem connects the idea of roots and factors together.

• A root of \(p(x)\) is a solution to the equation \(p(x) = 0\).
• A factor of \(p(x)\) is an expression that divides \(p(x)\) perfectly with no remainder.

By leveraging the Factor Theorem, you can simplify polynomial division problems and gain insight into the structure of the polynomial itself.

Synthetic Division

Synthetic Division is a simplified form of polynomial division, particularly effective when dividing by a linear polynomial like \(x - r\). It's a quicker, more streamlined approach than long division.

In synthetic division, you use only the coefficients of \(p(x)\) and the root, in your division. Here's a basic outline of how it works:

• List the coefficients of the polynomial in order.
• Write the root to the left and proceed through a series of multiplication and addition operations.
• Work your way across the coefficients, repeating the process until you reach the end.
• Check the remainder: for the root to be valid, the remainder should be zero.

This method is not only faster but also helps in validating if a specific \(q(x)\) is a factor of the polynomial.

Polynomials

Polynomials are expressions consisting of variables and coefficients organized in terms of powers, usually presented in descending order. Each term in a polynomial is a product of a constant and a variable raised to a non-negative integer power.

Polynomials can have one or more terms, like:\[x^4 - 50\] which is called a "quartic" polynomial because of its highest degree of 4. Understanding polynomials is fundamental to mastering concepts like Polynomial Division and the Factor Theorem.

• Terms: Each separate component of a polynomial.
• Coefficient: The number multiplying each term's variable.
• Degree: The highest power of \(x\) in the polynomial.

These structures help organize polynomial expressions, allowing for clear and concise mathematical reasoning and solutions.

Roots of Polynomials

Roots of polynomials are vital to understanding their behavior. A root is any value that, when substituted into the polynomial, results in a zero value.

Finding the roots of a polynomial involves determining the values that satisfy this zero condition. The roots are closely linked to factors: each root \(r\) provides a factor \((x - r)\).

• Roots can have different multiplicities, affecting the polynomial's graph at those points.
• Real roots result in the polynomial crossing or touching the x-axis.
• Complex roots come in pairs and, when plotted, create symmetrical features on a graph.

Understanding the roots of a polynomial is crucial for plotting graphs and can greatly simplify polynomial division, especially using the Factor Theorem or Synthetic Division.