Problem 35 - Intermediate Algebra

Question

Perform each division. \(\frac{t^{3}+8 t^{2}+13 t+9}{t+6}\)

Step-by-Step Solution

Answer

The division results in \(t^2 + 2t + 1 + \frac{3}{t + 6}\).

Step 1: Set up the Division

To divide the polynomial \(t^3 + 8t^2 + 13t + 9\) by \(t + 6\), we will use polynomial long division. Write the dividend \(t^3 + 8t^2 + 13t + 9\) and the divisor \(t + 6\) in long division form.

Step 2: Divide the First Term

Divide the first term of the dividend, \(t^3\), by the first term of the divisor, \(t\): \[ t^3 \div t = t^2 \]This result, \(t^2\), is the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(t^2\) by the entire divisor \(t + 6\): \[ t^2(t + 6) = t^3 + 6t^2 \]Subtract this product from the original polynomial: \[ (t^3 + 8t^2 + 13t + 9) - (t^3 + 6t^2) = 2t^2 + 13t + 9 \]

Step 4: Repeat the Process

Divide the first term of the new dividend, \(2t^2\), by the first term of the divisor, \(t\): \[ 2t^2 \div t = 2t \]Multiply \(2t\) by the divisor \(t + 6\): \[ 2t(t + 6) = 2t^2 + 12t \]Subtract this from the current dividend: \[ (2t^2 + 13t + 9) - (2t^2 + 12t) = t + 9 \]

Step 5: Final Division Step

Divide the first term of the current dividend, \(t\), by the first term of the divisor, \(t\): \[ t \div t = 1 \]Multiply \(1\) by the divisor \(t + 6\): \[ 1(t + 6) = t + 6 \]Subtract this from the current dividend to find the remainder: \[ (t + 9) - (t + 6) = 3 \]

Step 6: Write the Final Answer

Combine the results of the division. The quotient is \(t^2 + 2t + 1\) and the remainder is \(3\). Thus, the division can be expressed as:\[ \frac{t^3 + 8t^2 + 13t + 9}{t + 6} = t^2 + 2t + 1 + \frac{3}{t + 6} \]

Key Concepts

Polynomial Long Division

Polynomial long division is much like the long division process we use for numbers, but for these expressions. It is a method used to divide polynomials, allowing you to find both the quotient and remainder.

Here's how it works for our example,

• First, the dividend (what you are dividing) is written, which in our case is \(t^3 + 8t^2 + 13t + 9\).
• The divisor (what you are dividing by) is \(t + 6\).
• Then, using the long division method, divide the leading terms step-by-step until you've gone through all terms of the dividend.
• Each time you divide a term, multiply and subtract from the current dividend.

This process is repeated until you can no longer divide without getting a smaller degree polynomial term. The last number or polynomial left is the remainder. You can express the final answer as a quotient plus a fraction of the remainder over the divisor. In this exercise, that's: \( t^2 + 2t + 1 + \frac{3}{t+6}\).

Using polynomial long division is a valuable skill in algebra, especially when working with complex expressions.

Remainder Theorem

The remainder theorem is an important concept when working with polynomial division. It helps you to find the remainder of a division quickly without undergoing the full division process.

The theorem states: If a polynomial \(f(t)\) is divided by \(t - a\), then the remainder is \(f(a)\). So, to find the remainder when dividing a polynomial by a binomial, you simply substitute the zero of the binomial into the polynomial.

For the exercise at hand, if you were dividing the polynomial \(t^3 + 8t^2 + 13t + 9\) by \(t + 6\), the zero of the divisor \(t + 6\) is \(-6\). By substituting \(-6\) into the polynomial:
\[(-6)^3 + 8(-6)^2 + 13(-6) + 9 = 3\]
This confirms the remainder is 3, matching the remainder we found through polynomial long division. The remainder theorem simplifies checking the accuracy of your work.

Algebraic Expressions

Algebraic expressions are a staple of algebra and represent numbers using symbols or letters. This allows you to generalize mathematical ideas and calculations. They can include numbers, variables, and operations.

In our division example, \(t^3 + 8t^2 + 13t + 9\) and \(t + 6\) are algebraic expressions. Understanding how to manipulate these expressions through operations like addition, subtraction, and division is crucial.

• Variables like \(t\) act as placeholders and can change value, enabling the solving of equations with unknowns.
• Polynomials are a type of algebraic expression, characterized by having multiple terms which can either include constants or variables raised to a power.

This division exercise highlights the concept of how algebraic expressions can be divided and simplified, further enhancing the comprehension of relationships within equations. Proficiency in handling algebraic expressions lays the foundation for advanced math topics.