Problem 1 - Precalculus : Building Concepts and Connections

Question

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$2 x^{2}+13 x+15 ; x+5$$

Step-by-Step Solution

Answer

The quotient is \(2x + 3\) and the remainder is 0.

Step 1: Arrange the Polynomials

Write down the coefficients of the polynomials in the order of decreasing powers of \(x\). From the highest to lowest degree of the polynomial, the first polynomial's coefficients are 2, 13, and 15. The second polynomial's coefficients are 1 and 5.

Step 2: Use Synthetic Division

Begin synthetic division. In this case, -5 (opposite of +5 in the divisor \(x+5\)) is used as the synthetic division number. Write -5 to the left and the numbers obtained from the first polynomial (2, 13, 15) to the right.

Step 3: Execute Synthetic Division

Perform the operations of synthetic division. Bring down the first number of the dividend (2). Multiply this number by -5 to obtain -10, and add this to the next number from the dividend (13), which results in 3. Repeat these steps for any remaining numbers from the dividend.

Step 4: Interpret the Result

The last number in the last row (0 in this case) is the remainder. The numbers above form the coefficients of the quotient. Since we started with a second degree polynomial and divided it by a first degree polynomial, the result is a first degree polynomial. In this case, the result is \(2x + 3\), with 0 remainder.

Key Concepts

Polynomial Division

Polynomial division is similar to the long division you learned in elementary school, but instead of numbers, we're working with polynomials. It involves dividing one polynomial by another, usually resulting in a quotient and a remainder. Here’s a breakdown of the process:

• Divisions Term by Term: You divide the highest degree of the dividend by the highest degree of the divisor.
• Subtract: Multiply the result by the divisor and subtract from the dividend.
• Repeat: Continue the process with the result from the subtraction as a new dividend.

All polynomials can be divided using this method, providing much-needed solutions, especially when factoring polynomials or evaluating limits in calculus. When the remainder is zero, the division is exact, meaning the divisor is a factor of the dividend.

Quotient and Remainder

When performing polynomial division, the goal is to find two items: a quotient and a remainder. The equation that represents the division is:\[ P(x) = D(x) \cdot Q(x) + R(x) \]Where:

• \(P(x)\): Dividend, the polynomial you're dividing.
• \(D(x)\): Divisor, the polynomial you are dividing by.
• \(Q(x)\): Quotient, the result of the division.
• \(R(x)\): Remainder, what's left over.If the remainder is zero, the divisor is a factor of the dividend.

In our example, we divided \(2x^2 + 13x + 15\) by \(x + 5\). We found that the quotient was \(2x + 3\) and the remainder was 0.

Synthetic Substitution

Synthetic substitution is a shortened form of polynomial division. It is useful for dividing polynomials when the divisor is in the form \(x-c\). This method reduces the complexity by focusing solely on the coefficients.Here’s how it simplifies the process:

• Quick Set-Up: Arrange the coefficients of the dividend in descending order of powers. The number used is the opposite of the constant term in the divisor.
• Simple Execution: Perform the operations without variable calculations, working just with numbers.
• Find Remainder and Quotient: The final number in your synthetic grid is the remainder. The other numbers are the coefficients of your quotient.

In our exercise example, we used -5 for the synthetic division with dividend coefficients 2, 13, and 15. This process efficiently demonstrated that the quotient is \(2x + 3\) with a remainder of 0.