Problem 21 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(2 x^{3}+9 x^{2}-17 x+6\right) \div(2 x-1) $$

Step-by-Step Solution

Answer

The quotient is \(x^2 + 5x - 6\).

Step 1: Set Up the Division

We are using polynomial long division to divide \(2x^3 + 9x^2 - 17x + 6\) by \(2x - 1\). Write the dividend inside the division symbol and the divisor outside.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend, \(2x^3\), by the leading term of the divisor, \(2x\), to get \(x^2\). Write \(x^2\) above the division symbol.

Step 3: Multiply and Subtract

Multiply the whole divisor \(2x - 1\) by \(x^2\) to get \(2x^3 - x^2\). Subtract this result from the original polynomial, which gives a new polynomial: \(10x^2 - 17x + 6\).

Step 4: Repeat Division

Divide the new leading term \(10x^2\) by the leading term \(2x\) to get \(5x\). Write \(5x\) above the division symbol next to \(x^2\).

Step 5: Multiply and Subtract Again

Multiply \(2x - 1\) by \(5x\) to get \(10x^2 - 5x\). Subtract this from the new polynomial to get \(-12x + 6\).

Step 6: Continue with Division

Divide the term \(-12x\) by \(2x\) to get \(-6\). Write \(-6\) above the division symbol.

Step 7: Final Multiplication and Subtraction

Multiply \(2x - 1\) by \(-6\) to get \(-12x + 6\). Subtract this from \(-12x + 6\) to get a remainder of 0.

Step 8: Write the Final Answer

Since the remainder is 0, the result of the division is \(x^2 + 5x - 6\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to how you divide numbers. It's handy when you have to simplify expressions or solve polynomial equations.

To start, you'll arrange the polynomials in a specific format:

• Place the dividend (the polynomial you are dividing) inside the division symbol.
• Place the divisor (the polynomial you are dividing by) outside the division symbol.

For example, in the expression \((2x^3 + 9x^2 - 17x + 6) \div (2x - 1)\), the dividend is \(2x^3 + 9x^2 - 17x + 6\) and the divisor is \(2x - 1\).

To solve by polynomial long division, you'll follow these steps:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this result and subtract it from the dividend.
• Bring down the next term from the original dividend and repeat the process.
• Continue until you reach a remainder of zero or a polynomial of a lesser degree than the divisor.

This method helps to transform complex polynomial expressions into simpler ones, which is essential in higher-level algebra.

Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that connects polynomial division with evaluating polynomials. It states that when a polynomial \(f(x)\) is divided by a linear divisor of the form \((x - a)\), the remainder of this division is simply \(f(a)\).

This theorem is incredibly efficient because:

• You can quickly find the remainder without performing the full division.
• It's a fast check for factorization, as if \(f(a) = 0\), then \((x-a)\) is a factor of the polynomial.

In our earlier division problem, the remainder was zero. This tells us that \(2x - 1\) is a perfect divisor of \(2x^3 + 9x^2 - 17x + 6\) and confirms that the operation was successful.

This theorem simplifies solving polynomial equations and is part of what makes algebraic manipulation powerful.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and arithmetic operations such as addition, subtraction, multiplication, and division. They form the basis of algebraic equations and inequalities.

Polynomials are a type of algebraic expression that includes terms with non-negative integer exponents. The expression \(2x^3 + 9x^2 - 17x + 6\) is a polynomial. Here, different terms are organized based on their degree, which is the exponent of the variable.

Understanding algebraic expressions is crucial because:

• It allows you to model and solve real-world problems.
• You can simplify and rearrange these expressions to make calculations easier or to find unknown values.

By mastering polynomial manipulation, such as adding, subtracting, multiplying, and dividing, you build a strong foundation for advancing in math topics like calculus and beyond.
Algebraic expressions are versatile tools that lead to a deeper understanding of mathematical relationships and structures.