Problem 27 - Precalculus Mathematics for Calculus

Question

Find the quotient and remainder using synthetic division. $$\frac{3 x^{2}+5 x}{x-6}$$

Step-by-Step Solution

Answer

The quotient is \(3x + 23\) with a remainder of 23.

Step 1: Identify coefficients and the divisor

For synthetic division, take the coefficients of the polynomial and the zero of the divisor. Here, the coefficients are 3 and 5 from the polynomial \(3x^2 + 5x\). The root of the divisor \(x - 6\) is 6.

Step 2: Set up the synthetic division

Write down the zero of the divisor (6) on the left. Write the coefficients of the polynomial (3 and 5) to the right of this number. If there's a missing term, you would add a 0 for its coefficient, but here there isn't one.

Step 3: Bring down the first coefficient

Take the first coefficient, which is 3, and bring it down. This starts the synthetic division process.

Step 4: Multiply and add

Multiply the root (6) by the number just written below the line (3) and place the product (18) under the next coefficient (5). Add the 5 and 18 to get 23. This sum represents the next coefficient in the quotient.

Step 5: Write the result

The quotient becomes \(3x + 23\), and since there are no more coefficients to process, 23 is your remainder.

Key Concepts

Polynomial Division

When we deal with polynomial division, we're taking a large polynomial and dividing it by a simpler polynomial. This process is similar to long division but involves polynomials instead of numbers. In our example, we have the polynomial dividend, which is given as \(3x^2 + 5x\), and the divisor \(x - 6\).

• The goal is to simplify the expression \(\frac{3x^2 + 5x}{x-6}\) by using polynomial division.
• This allows us to express it in the form of a quotient (the result of division) and a remainder (what's left over).

This technique can help solve polynomial equations and simplify complex polynomial expressions.

Remainder Theorem

The Remainder Theorem is a handy tool in polynomial mathematics that states that the remainder of the division of a polynomial \(f(x)\) by a linear polynomial \(x - a\) is \(f(a)\).
In our exercise, we apply this theorem when we find that the remainder is 23 after dividing the polynomial \(3x^2 + 5x\) by \(x - 6\).

• The theorem gives us a quick check: if we substitute \(x = 6\) (the zero of the divisor) into the polynomial and calculate, the result should be equal to the remainder from synthetic division.
• This theorem is powerful because it not only checks division results but also helps in determining factors of the polynomial. If f(a) is zero, it verifies that \(x - a\) is a factor of the polynomial \(f(x)\).

With the remainder theorem, understanding the divisor's role in polynomial expressions becomes much clearer.

Coefficients

Coefficients in a polynomial are essential elements that define the polynomial's terms and shape. For the polynomial \(3x^2 + 5x\), 3 and 5 are the coefficients.

• 3 is the coefficient of the term \(x^2\), indicating three times the square of \(x\).
• 5 is the coefficient of the term \(x\), representing five times \(x\).

During synthetic division, we primarily focus on these coefficients without worrying about the higher powers of \(x\).
This simplifies the math since we only perform operations with these numbers. Coefficients are crucial in determining the polynomial's behavior and assist in various algebraic manipulations like solving equations and factoring.

Division Algorithm

The Division Algorithm for polynomials is a formula that describes a way to divide one polynomial by another. The division process results in a quotient and a remainder.
For our given problem, the division algorithm states the division can be expressed as:\[P(x) = (D(x) \cdot Q(x)) + R(x)\]where:

• \(P(x)\) is \(3x^2 + 5x\), the dividend.
• \(D(x)\) is \(x - 6\), the divisor.
• \(Q(x)\) is \(3x + 23\), the quotient.
• \(R(x)\) is 23, the remainder.

This relationship shows how each element fits into the division process, and understanding this can help verify the results.
The division algorithm is foundational in determining how polynomials behave under division and is vital for more advanced algebraic concepts.