Problem 94 - Introductory Algebra for College Students

Question

Divide \(2 x^{3}+x^{2}-13 x+6\) by \(x-2\) Use the quotient to factor \(2 x^{3}+x^{2}-13 x+6\) completely.

Step-by-Step Solution

Answer

The polynomial \(2x^3 + x^2 -13x + 6\) can be factored completely as \((x-2)(2x^2 + 5x + 1)\).

Step 1: Polynomial Division

For this, the divisor \(x-2\) is to be divided from the polynomial \(2x^3 + x^2 -13x +6\). Start by dividing the highest degree term in the dividend, \(2x^3\), by the highest degree term in the divisor, \(x\), which results in \(2x^2\). Then, multiply \(2x^2\) by the divisor \(x-2\) to get \(2x^3 - 4x^2\), and subtract that from the original dividend to get \(-3x^2 - 13x + 6\). Continuing the process until the degree of remainder is less than the degree of divisor, results in the quotient \(2x^2 + 5x + 1\) and a remainder of 0.

Step 2: Factoring the Polynomial

The original polynomial \(2x^3 + x^2 -13x + 6\) can now be factored completely by using the divisor and the quotient from the polynomial division step. It can be expressed as the product of \(x-2\) and \(2x^2 + 5x + 1\). This can be written as \((x-2)(2x^2 + 5x + 1)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra, allowing us to break down complex expressions into simpler components. A polynomial is composed of variables, coefficients, and exponents combined through addition, subtraction, and multiplication. When factoring, the goal is to express a given polynomial as a product of its factors—simpler polynomials or terms. To factor a polynomial, one common technique is utilizing polynomial division to find its factors. Once a factor is identified, like a divisor with a zero remainder, it can be used to further break down the polynomial until it is fully decomposed into irreducible factors. In our case, the polynomial division yielded the factors \((x-2)\) and \((2x^2 + 5x + 1)\). Hence, \(2x^3 + x^2 - 13x + 6\) can be expressed as:

• \((x-2)(2x^2 + 5x + 1)\).

This expression shows the two factors whose product results in the original polynomial. Fully understanding and performing polynomial factoring is useful in solving equations and modeling real-world problems.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of numbers, variables, and mathematical operations. Understanding algebraic expressions is essential because they represent quantities and relationships using symbols. An algebraic expression, like the polynomial in our example, can often seem complicated. However, it can be simplified or manipulated in many ways, including factoring or expanding through operations. When dealing with polynomials, you might see expressions like \(2x^3 + x^2 - 13x + 6\), which are combinations of terms where each term is a product of a constant and a variable raised to a power. Recognizing how to transform these expressions either by combining like terms, simplifying, or breaking them down is a key skill in algebra. Practice helps make evaluating, simplifying, and operating with algebraic expressions more intuitive. This simplifies complex problems and aids in solving equations efficiently.

Long Division in Algebra

Long division in algebra is an extension of the traditional long division used with numbers. It is used to divide polynomials, helping find quotients and remainders when one polynomial is divided by another. In our example, long division was used to divide \(2x^3 + x^2 - 13x + 6\) by \(x-2\). The process begins by dividing the leading term of the dividend by the leading term of the divisor. This division gives the first part of the quotient. Next, multiply this term by the entire divisor and subtract the result from the original polynomial. This process is repeated with each resulting polynomial until the remainder is of a lower degree than the divisor. The division process adheres to these steps:

• Find the first term of the quotient.
• Multiply and subtract to form the new dividend.
• Repeat until a remainder is obtained, if any.

Through long division, the quotient \(2x^2 + 5x + 1\) was found, signifying our full understanding of the division process while leading us to successfully factor the polynomial.