Problem 9 - Intermediate Algebra

Question

Fill in the blanks. Rather than substituting 8 for \(x\) in \(P(x)=6 x^{3}-x^{2}-17 x+9\) we can divide the polynomial _____ by ____ to find \(P(8)\).

Step-by-Step Solution

Answer

Divide the polynomial \(6x^3 - x^2 - 17x + 9\) by \(x - 8\) to find \(P(8)\).

Step 1: Identify the Operation

The problem suggests using polynomial division to find \(P(8)\). Rather than directly substituting \(x = 8\) into the polynomial function, we will perform synthetic or long division.

Step 2: Determine the Needed Division

To find \(P(8)\) by division, we need to divide the polynomial \(P(x) = 6x^3 - x^2 - 17x + 9\) by \(x - 8\). This is because evaluating \(P(8)\) using division corresponds to finding the remainder when the polynomial is divided by \(x - c\), where \(c = 8\).

Step 3: Select the Terms

The blanks to fill in the question are: 1. The polynomial which is \(P(x)\) or \(6x^3 - x^2 - 17x + 9\).2. The divisor which is \(x - 8\), since we are finding \(P(8)\) using division.

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial division, which is particularly useful for dividing a polynomial by a binomial of the form \(x - c\). It provides a quicker and more efficient way to get the same results as long division.

Here’s how you can perform synthetic division:

• Write down the coefficients of the polynomial in standard form. For the polynomial \(6x^3 - x^2 - 17x + 9\), the coefficients are 6, -1, -17, and 9.
• Use the root of the divisor \(x - 8\), which is 8, to perform the division.
• Bring down the leading coefficient (6 in this case) to the bottom row.
• Multiply this leading coefficient by 8, and write the result under the next coefficient (-1).
• Add the numbers in the column to get the new number beneath.
• Repeat the process of multiplying and adding until you reach the last coefficient.
• The last number you get is the remainder, which is actually the value of \(P(8)\).

Using synthetic division, not only do you find the remainder quickly, but you also determine the quotient of the division. This method considerably reduces the amount of written work compared to traditional long division.

Remainder Theorem

The remainder theorem is a key concept in polynomial algebra. It states that if a polynomial \(P(x)\) is divided by \(x - c\), the remainder of this division is \(P(c)\). Thus, rather than substituting the value directly into the polynomial, division provides the same value.

For example, when we divide the polynomial \(P(x) = 6x^3 - x^2 - 17x + 9\) by \(x - 8\), the remainder theorem tells us that the remainder is precisely the value \(P(8)\). This technique is particularly useful for evaluating polynomial functions at specific points while avoiding potentially tedious arithmetic.

The remainder theorem not only allows us to evaluate polynomials efficiently but also offers insights into factorization and root-finding. If the remainder is zero, \(x - c\) is a factor of the polynomial, indicating \(c\) is a root.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it’s an algebraic expression like \(6x^3 - x^2 - 17x + 9\) where \(x\) is the variable, and the numbers are coefficients.

Here are some characteristics of polynomial functions:

• They can have coefficients (numbers in front of \(x\)) that are constants.
• The exponents must be positive integers. For example, \(x^3\) is valid, but \(x^{-2}\) is not.
• Polynomials are continuous and have smooth curves, making them predictable and easier to graph than other types of functions.

Polynomial functions can vary from simple ones like \(3x + 2\) to more complex expressions with higher degrees like cubic functions or higher. Understanding them is essential, as they form the foundation for more advanced mathematical topics.

Long Division

Long division is the more traditional and comprehensive method of dividing two polynomials. It resembles the long division algorithm used with numbers but applies to polynomial expressions. This method includes dividing, multiplying, and subtracting steps repeatedly until the remainder is less than the divisor.

To use long division on a polynomial like \(6x^3 - x^2 - 17x + 9\) divided by \(x - 8\), you would:

• Divide the leading term of the dividend by the leading term of the divisor, writing the result over the dividend.
• Multiply the entire divisor by this result and subtract from the dividend.
• Bring down the next term from the dividend and repeat these steps until you reach the final remainder.

Long division is useful for simple and complex polynomials, providing a systematic approach to understanding divisions thoroughly even when synthetic division isn't applicable. It’s particularly useful when dividing by polynomials that are not binomials of the form \(x - c\).