Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x-2, \quad 3 x^{4}-6 x^{3}-5 x+10 $$

Step-by-Step Solution

Answer

\(x - 2\) is a factor. Factorization: \( 3x^3 - x^2 - 10 \).

Step 1: Set up the Synthetic Division

Identify the divisor from the expression \(x - 2\), which is 2. Write it down and arrange the coefficients of the dividend \(3x^4 - 6x^3 + 0x^2 - 5x + 10\) in a row: \([3, -6, 0, -5, 10]\).

Step 2: Begin Synthetic Division Calculation

Bring down the first coefficient (3) as it is. Multiply this number by 2 (the divisor) and write the result below the second coefficient. Add this result to the second coefficient (-6) and write the sum under the same column.

Step 3: Continue the Division

Repeat the process: multiply the result from Step 2 by 2, write it under the next coefficient, then add down. Continue this until you reach the last coefficient.

Step 4: Evaluate the Remainder

After completing the division process, look at the final number in the bottom row, which is the remainder. If this remainder is zero, \(x-2\) is a factor; otherwise, it is not.

Step 5: Identify the Resultant Polynomial

If \(x-2\) is a factor (remainder is zero), write out the resulting polynomial from the coefficients in the bottom row, excluding the remainder.

Key Concepts

Polynomial Division

Polynomial division is a fundamental concept in algebra, akin to long division with numbers. It allows us to divide one polynomial by another, reducing complex expressions. The process usually involves dividing a polynomial by a monomial or another polynomial of lower degree. Synthetic division, a simplified form of polynomial division, is especially handy when dividing by a linear polynomial with leading coefficient 1, such as \(x - a\). It reduces the effort and potential for errors by limiting operations to just the coefficients.To perform synthetic division:

• Identify the divisor and arrange coefficients of the dividend.
• Perform a series of straightforward multiplications and additions.

This approach not only checks divisibility but also helps us deduce the quotient and any remainder. Understanding this method is crucial for solving polynomial equations effectively.

Remainder Theorem

The remainder theorem is a key concept when working with polynomial division, as it relates directly to synthetic division. It states that the remainder of the division of a polynomial \(f(x)\) by a linear divisor \(x-a\) is simply \(f(a)\). This theorem simplifies finding the remainder without completing the entire division process. By substituting \(a\) into the polynomial, you instantly know the remainder.Applying this theorem:

• Calculate \(f(a)\) where \(a\) is the root of the divisor.
• Evaluate whether the result is zero (indicating the divisor is a factor).

This principle not only streamlines calculations but aids in verifying potential polynomial factors swiftly.

Factorization

Factorization is the process of expressing a polynomial as a product of its factors. Understanding how to factor polynomials is essential in algebra, as it can simplify equations and help solve polynomial equations. If after performing synthetic division the remainder is zero, it's confirmed that the divisor is a factor of the polynomial. The steps in factorization include:

• Perform synthetic division checking for zero remainders.
• Express the polynomial as a product of the quotient and divisor.
• Simplify further if possible by factoring the quotient.

This method breaks down complex expressions into simpler, manageable parts, making problem-solving much more approachable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators representing specific values. Understanding algebraic expressions is fundamental to mastering algebra and calculus. They provide a powerful way to describe relationships and changes in values comprehensively. The composition of algebraic expressions includes:

• Terms - individual parts of the expression separated by addition or subtraction.
• Coefficients - numbers multiplying the variables in the terms.
• Variables - symbols representing unknown or changeable values.

Grasping the structure of algebraic expressions enables easy manipulation and transformation of formulas, leading to effective problem-solving and deeper mathematical understanding.