Problem 39 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The quotient is \(2x - 1\).

Step 1: Setup Long Division

Start by setting up the polynomial long division. Write the dividend \(2x^3 + x^2 - 3x + 1\) under the division symbol and the divisor \(x^2 + x - 1\) outside of it.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x^2\). This gives \(2x\). Write \(2x\) above the division line.

Step 3: Multiply and Subtract

Multiply \(2x\) by the entire divisor \(x^2 + x - 1\), resulting in \(2x^3 + 2x^2 - 2x\). Subtract this from the original dividend \(2x^3 + x^2 - 3x + 1\), which leaves \(-x^2 - x + 1\).

Step 4: Repeat Process

Divide the new leading term \(-x^2\) by the leading term \(x^2\) of the divisor, resulting in \(-1\). Write \(-1\) above the division line next to \(2x\).

Step 5: Multiply and Subtract Again

Multiply \(-1\) by the whole divisor \(x^2 + x - 1\), which results in \(-x^2 - x + 1\). Subtract this result from \(-x^2 - x + 1\), giving a zero remainder indicating that the division is exact.

Step 6: Write Down the Quotient

The quotient obtained from this division is the expression written above the division line, which is \(2x - 1\). This is the solution.

Key Concepts

Polynomial Division

Polynomial division is similar to dividing numbers but involves algebraic expressions. It's a way to divide one polynomial by another, resulting in a quotient and possibly a remainder. The process starts by writing the dividend, which is the polynomial you're dividing, under a division bracket. The divisor goes outside. The goal is to simplify the polynomial expression by determining how many times the divisor can fit into it.

To perform the division, follow a systematic process:

• Identify the leading term of the dividend and compare it with the leading term of the divisor.
• Divide these leading terms to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the dividend.
• Bring down the next term of the dividend if necessary and repeat the steps until all terms are processed.

This method allows us to simplify complex polynomials into more manageable parts, which is fundamental in algebra for solving equations and simplifying expressions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Similar to numerical fractions, it's possible to simplify, add, subtract, multiply, and divide these expressions. Therefore, it's essential to understand how polynomial division plays a role in these processes.

During polynomial division, the division process can transform complex rational expressions into simpler forms. This is particularly useful when the rational expression is a part of a larger algebraic equation. Ensuring the simplified form is crucial for solving these equations effectively.

Remember:

• A zero remainder in polynomial division indicates that the original expression can exactly be divided, meaning the divisor is a factor of the dividend.
• If any terms remain, they become a part of the remainder, similar to how remainders work in arithmetic division.

Mastering rational expressions requires practice, focusing on combining like terms and managing both numerators and denominators efficiently.

Algebraic Operations

Algebraic operations extend beyond simple arithmetic operations to include manipulation of polynomials and rational expressions. These operations involve addition, subtraction, multiplication, division, and factoring of algebraic terms.

Considerations when dividing polynomials include checking for common factors and using techniques like polynomial long division or synthetic division, when applicable, to streamline the process. Understanding the order of operations ensures accuracy in solving algebraic equations.

When working with algebraic operations:

• Ensure terms are correctly aligned when adding or subtracting polynomials.
• Apply the distributive property accurately in multiplication.
• For division, remember to divide and then turn subtraction of terms as explained in polynomial division.

Proficiency in algebraic operations ensures that complex expressions can be simplified and solved efficiently, a cornerstone skill in more advanced mathematics.