Problem 26 - A Graphical Approach to College Algebra

Question

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=2 x^{4}+3 x^{3}-5 x^{2}-18 x ; \quad x-2$$

Step-by-Step Solution

Answer

The quotient is \(2x^3 + 7x^2 + 9x\).

Step 1: Identify the dividend and the divisor

The given polynomial is \(P(x) = 2x^4 + 3x^3 - 5x^2 - 18x\), which is our dividend. The divisor is the binomial \(x - 2\). We need to find the quotient of this division.

Step 2: Set up synthetic division

Since we are dividing by \(x - 2\), we will use synthetic division and place \(2\) to the left of the division table. The coefficients of the dividend \(2x^4 + 3x^3 - 5x^2 - 18x\) are \([2, 3, -5, -18, 0]\). The zero at the end is a placeholder for the \(x^0\) term.

Step 3: Perform synthetic division

Write down the first coefficient (2) below the line. Multiply the number just written (2) by 2 (the value from \(x - 2\)) and write the result (4) in the next column above the line. Add the number in the next column (3) to the result (4), giving 7. Repeat this process: multiply 7 by 2 resulting in 14, add to \(-5\) to get 9, multiply 9 by 2 to get 18, add to \(-18\) to get 0. The last multiplication results in 0, which is the remainder.

Step 4: Write the quotient

The quotient is represented by the coefficients obtained from synthetic division: \(2, 7, 9, 0\). Thus, the quotient polynomial is \(2x^3 + 7x^2 + 9x\). There is no remainder.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. It is similar to numerical long division you may have learned in elementary school, but here we deal with variables and coefficients. When you perform polynomial division, you essentially break down the dividend polynomial into terms that are each divisible by the divisor polynomial. This helps to simplify complex expressions, making them easier to manage and solve.

In the exercise, the dividend, or the polynomial being divided, is \(P(x) = 2x^4 + 3x^3 - 5x^2 - 18x\). The divisor is a simpler polynomial, a binomial, \(x - 2\). Our goal is to determine how many times this divisor can "fit" into the dividend. This results in a quotient, which represents the polynomial that you've divided, plus any remainder, if applicable.

Key points to remember about polynomial division:

• The dividend is the polynomial you are dividing.
• The divisor is the polynomial you divide by.
• The process can result in a quotient and sometimes a remainder.
• Understanding this process lays groundwork for methods like synthetic division.

Remainder Theorem

The Remainder Theorem is a useful concept when working with polynomials. It provides a quick way to find the remainder of a division operation when a polynomial \(P(x)\) is divided by a linear divisor of the form \(x - c\). According to the theorem, if you divide a polynomial \(P(x)\) by \(x - c\), the remainder of this division is simply \(P(c)\).

In our exercise, the divisor is \(x - 2\). According to the Remainder Theorem, if we substitute 2 into the polynomial \(P(x) = 2x^4 + 3x^3 - 5x^2 - 18x\), the result should be the remainder. Performing synthetic division gave us a remainder of 0, which means \(P(2) = 0\). This shows that \(x - 2\) is a factor of \(P(x)\).

Why the Remainder Theorem matters:

• It provides a quick check for divisibility.
• You can use it to verify division operations and synthetic division results.
• Helps in determining if a binomial is a factor of a polynomial.

Quotient Polynomial

In synthetic division, finding the quotient polynomial is the primary objective. The quotient is the polynomial that results from dividing the original polynomial by the divisor.

In the exercise above, after performing synthetic division, the quotient polynomial is found to be \(2x^3 + 7x^2 + 9x\). These terms are derived from the coefficients generated during the synthetic division process. The coefficients \([2, 7, 9, 0]\) directly translate into the terms of the quotient polynomial.

Some things to consider about quotients:

• The degree of the quotient polynomial is one less than the degree of the dividend.
• The quotient, together with the remainder, perfectly represents the division of the original polynomial \(P(x)\).
• Understanding quotients helps simplify and solve equations or expressions.

Knowing the quotient can often provide insight into solving polynomial equations, factoring, or further simplifying algebraic expressions. It is an integral part of understanding polynomial behavior.