Problem 10 - Basic Engineering Mathematics

Question

Divide \(2 x^{2}+x-3\) by \(x-1\)

Step-by-Step Solution

Answer

The quotient is \(2x + 3\).

Step 1: Set up the division

Write the polynomial division expression as \((2x^2 + x - 3) \div (x - 1)\). This means we are dividing \(2x^2 + x - 3\) by \(x - 1\).

Step 2: Divide the leading term

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

Step 3: Multiply and subtract

Multiply \(2x\) by \(x - 1\), resulting in \(2x^2 - 2x\). Subtract this from the original polynomial: \((2x^2 + x - 3) - (2x^2 - 2x) = 3x - 3\).

Step 4: Divide the new leading term

Take the new leading term \(3x\) and divide it by \(x\), which gives us \(3\). Write \(3\) above the division bracket, next to \(2x\).

Step 5: Multiply and subtract again

Multiply \(3\) by \(x - 1\), which results in \(3x - 3\). Subtract \(3x - 3\) from the remaining polynomial \(3x - 3\), leaving a remainder of \(0\).

Step 6: Write the quotient

Since the remainder is \(0\), the quotient is \(2x + 3\), which means \(2x^2 + x - 3 = (x - 1)(2x + 3)\).

Key Concepts

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form \(x - c\). It's a quick alternative to traditional long division, especially when dealing with linear divisors. To perform synthetic division, you first set up the synthetic division table. This involves writing the coefficients of the dividend polynomial in a row. For example, for \(2x^2 + x - 3\), you would write \([2, 1, -3]\). The divisor \(x - 1\) gives us \(c = 1\) to use in our table.

• Drop down the leading coefficient directly as it is - in this case, the 2.
• Multiply this by \(c\) (here, \(1\)), and write the result underneath the next coefficient.
• Add these two numbers vertically, and repeat the process until you reach the end of the row.

With synthetic division, the final row consists of the coefficients of your quotient, and any remainder if there is one. This method saves time and simplifies calculations, especially for those new to polynomial division.

Quotient Rule

In polynomial division, the term quotient refers to the result you obtain when dividing one polynomial by another. The quotient rule, in the context of polynomial division, differs from calculus, where it applies to differentiating functions. Here, the rule is all about organizing terms to systematically figure out which terms multiply to give you the original polynomial.
When dividing \(2x^2 + x - 3\) by \(x - 1\), the quotient is found using basic algebraic operations:

• First, divide the leading term of the dividend (the polynomial being divided) by the leading term of the divisor. This gives you the first part of the quotient.
• Multiply the resulting term by the entire divisor, and subtract it from the original polynomial. This is akin to peeling away layers, simplifying the polynomial step-by-step.
• Repeat the sequence with the new polynomial created by the subtraction.

Following these steps precisely allows you to determine the full quotient without error. Properly grasping this concept helps in solving complex divisions more efficiently.

Algebraic Expressions

Algebraic expressions are combinations of terms made up using numbers, variables, and operations like addition, subtraction, multiplication, and division. When we talk about dividing polynomials, we're simplifying these expressions by breaking them into smaller, more manageable parts.
Consider the expression \(2x^2 + x - 3\). It consists of three terms made of coefficients and the variable \(x\). Dividing this expression by another, like \(x - 1\), involves finding a simpler expression (quotient) that, when multiplied by the divisor, returns the original expression.
Understanding these expressions is crucial because:

• They often represent real-world situations, from calculating areas to predicting outcomes in finance.
• Manipulating them through operations such as division can simplify complex problems and provide clearer insights.

Algebraic expressions are foundational in algebra, and mastering them is a stepping stone to more advanced mathematical concepts.