Problem 45 - Intermediate Algebra

Question

Perform each division. \(\frac{13 x+16 x^{4}+3 x^{2}+3}{3+4 x}\)

Step-by-Step Solution

Answer

The result of the division is \(4x^3 - 3x^2 + 3x + 1\).

Step 1: Arrange the Terms

Make sure both the dividend and the divisor are arranged in descending order of the powers of x. The dividend is already presented in descending order as \(16x^4 + 3x^2 + 13x + 3\) and the divisor is \(4x + 3\). There are no terms missing in the dividend that have skipped powers of x.

Step 2: Divide the Highest Degree Terms

Divide the leading term of the dividend by the leading term of the divisor: \( \frac{16x^4}{4x} = 4x^3 \). This is the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(4x^3\) by the entire divisor \(4x + 3\) to get \(16x^4 + 12x^3\). Subtract this result from the original dividend: \(16x^4 + 3x^2 + 13x + 3 - (16x^4 + 12x^3) = -12x^3 + 3x^2 + 13x + 3\).

Step 4: Repeat the Division Process

Take the new dividend \(-12x^3 + 3x^2 + 13x + 3\) and divide \(-12x^3\) by the leading term of the divisor: \( \frac{-12x^3}{4x} = -3x^2 \). Add \(-3x^2\) to the quotient.

Step 5: Multiply and Subtract Again

Multiply \(-3x^2\) by \(4x + 3\) to get \(-12x^3 - 9x^2\). Subtract this from \(-12x^3 + 3x^2 + 13x + 3\): \(-12x^3 + 3x^2 + 13x + 3 - (-12x^3 - 9x^2) = 12x^2 + 13x + 3\).

Step 6: Continue the Process

With the new dividend \(12x^2 + 13x + 3\), divide \(12x^2\) by \(4x\): \( \frac{12x^2}{4x} = 3x \). Add \(3x\) to the quotient.

Step 7: Multiply and Subtract Yet Again

Multiply \(3x\) by \(4x + 3\) to get \(12x^2 + 9x\). Subtract from \(12x^2 + 13x + 3\): \(12x^2 + 13x + 3 - (12x^2 + 9x) = 4x + 3\).

Step 8: Final Division Step

Divide \(4x\) by \(4x\) to get \(1\), which is added to the quotient. Multiply \(1\) by \(4x + 3\) to get \(4x + 3\). Subtract from \(4x + 3\) yielding \(0\).

Step 9: Write the Final Answer

There is no remainder in this division process, so the division ends here. The polynomial result of the division is the quotient found through these steps: \(4x^3 - 3x^2 + 3x + 1\).

Key Concepts

Long Division

Polynomial long division is similar to the long division process you might remember from dividing numbers. Here, instead of dividing numbers, we divide polynomial expressions. Breaking down the steps can make the concepts easier to follow:

• Identify the Dividend and Divisor: The expression being divided is the dividend, and what you are dividing by is the divisor. In this case, the dividend is \(16x^4 + 3x^2 + 13x + 3\) and the divisor is \(4x + 3\).
• Arrange Terms Properly: Always write the terms in descending order of their powers. This helps maintain accuracy in the division process.
• Follow Step-by-Step: Start by dividing the first term of the dividend by the first term of the divisor, multiply the quotient term by the entire divisor, subtract, and repeat until the remainder is zero or a lower degree than the divisor.

The process requires patience and practice to master but becomes intuitive the more it's used.

Polynomial Expressions

A polynomial expression is a mathematical expression involving a sum of powers of a variable, each multiplied by a coefficient. For example, \(16x^4 + 3x^2 + 13x + 3\) is a polynomial with four terms:

• The term \(16x^4\) has a degree of 4.
• The term \(3x^2\) has a degree of 2.
• The term \(13x\) has a degree of 1.
• The constant term \(3\) has a degree of 0.

Polynomials are widely used in mathematics to represent equations and functions, making them crucial in algebra, calculus, and beyond. When working with polynomials, lining up terms by their degree helps simplify processes such as addition, subtraction, and division.

Quotient

The quotient is the result you obtain after dividing one expression by another. In polynomial division, the quotient is also a polynomial itself. From our example, dividing \(16x^4 + 3x^2 + 13x + 3\) by \(4x + 3\) gives the quotient \(4x^3 - 3x^2 + 3x + 1\). Each coefficient in the quotient corresponds to a degree of the divisor, shrinking step-by-step as you move down each division stage.

• The first term \(4x^3\) comes from dividing \(16x^4\) by \(4x\).
• Subsequent terms \(-3x^2\), \(3x\), and \(1\) result from the continuous process of division, multiplication, and subtraction as each step refines the remainder until nothing is left.

Understanding this concept helps recognize how divisions of polynomials often simplify complex math problems into manageable pieces, giving a foundational method to solving wider algebraic equations.