Problem 12 - Intermediate Algebra

Question

A polynomial is divided by \(3 a-2 .\) The quotient is \(3 a^{2}+5\) with a remainder of \(-6 .\) Write the answer to the division in two ways.

Step-by-Step Solution

Answer

The division of the polynomial gives: \( P(a) = 3a^3 - 6a^2 + 15a - 10 - 6\).

Step 1: Understand the Polynomial Division

In polynomial long division, when a polynomial \(P(a)\) is divided by a divisor \(D(a)\), it yields a quotient \(Q(a)\) and a remainder \(R\). Mathematically, this is expressed as \(P(a) = (D(a) \times Q(a)) + R\). Given: \(D(a) = 3a - 2\), \(Q(a) = 3a^2 + 5\), and \(R = -6\).

Step 2: Set Up the Equation from the Problem

Based on the polynomial division identity, substitute the known values into the equation. We have:\[P(a) = ((3a - 2) \times (3a^2 + 5)) - 6\]

Step 3: Perform the Multiplication

Multiply the divisor \(3a - 2\) by the quotient \(3a^2 + 5\) using the distributive property:1. Multiply \(3a\) by each term in \(3a^2 + 5\): - \(3a \times 3a^2 = 9a^3\) - \(3a \times 5 = 15a\)2. Multiply \(-2\) by each term in \(3a^2 + 5\): - \(-2 \times 3a^2 = -6a^2\) - \(-2 \times 5 = -10\)

Key Concepts

Quotient and Remainder

When dividing polynomials, just like dividing numbers, we end up with a quotient and possibly a remainder.

• The quotient is the result of the division.
• The remainder is what's left over if the polynomial doesn't divide evenly.

When you perform polynomial division, the goal is to express the original polynomial \(P(a) \) as the sum of two parts: the product of the divisor and the quotient, plus any remainder left over:

\[ P(a) = (D(a) \times Q(a)) + R \]For example, in our original exercise, dividing by \(3a-2\) resulted in a quotient of \(3a^2 + 5\) and a remainder of \(-6\). This means:

\[ P(a) = ((3a-2) \times (3a^2 + 5)) + (-6) \]This way, we can understand the relationship between the different components of polynomial division. By evaluating these pieces, much like taking apart a puzzle, it becomes easier to understand the entire expression.

Distributive Property

The distributive property is a key concept that helps ease polynomial multiplication.
When multiplying two polynomials, each term in the first polynomial is multiplied by every term in the second polynomial.

• If you think of it like a hand shake--each term shakes hands with every other term.
• This ensures all combinations of terms are multiplied together.

In our solution, we multiplied \(3a - 2\) by \(3a^2 + 5\) using this property:

• First, \(3a\) is multiplied by both \(3a^2\) and \(5\). It gives us two terms, \(9a^3\) and \(15a\).
• Then, \(-2\) is multiplied by both \(3a^2\) and \(5\), resulting in \(-6a^2\) and \(-10\).

Finally, we combine all these terms to build one complete polynomial expression. The distributive property, when applied correctly, ensures that no term is left out in the multiplication process.

Polynomial Multiplication

Polynomial multiplication involves increasing the order of the polynomial by performing term by term multiplication.
It's a step by step process where each term from the first polynomial is multiplied with every term from the second polynomial.

• This multiplication augments each term in terms of both its degree and its coefficient.
• During the process, terms involving similar powers can be combined to simplify the resulting expression.

For instance, in our exercise, multiplying the divisor \(3a - 2\) by the quotient \(3a^2 + 5\) resulted in several terms:

• \(9a^3\)
• \(-6a^2\)
• \(15a\)
• \(-10\)

Once combined, these terms paint a clearer picture of what the multiplication yields.
Understanding how polynomial multiplication expands expressions helps in solving complex algebraic expressions, offering insights into creating simplified, recognizable forms of larger polynomial expressions.