Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Divide using long division. $$\left(2 x^{4}+x^{2}-11\right) \div\left(x^{2}+5\right)$$

Step-by-Step Solution

Answer

The solution to \((2 x^{4}+x^{2}-11) \div (x^{2}+5)\) is \(2x^2 - 9\) with a remainder of 34.

Step 1: Arrange in Descending Order of Degree

First, arrange the terms of the dividend \(2 x^{4}+x^{2}-11\) and the divisor \(x^{2}+5\) in descending order of their degree. This means, the highest power of the variable comes first, followed by the next highest, and so on, until the term with no variable (the constant term) is listed last. In this case, both the dividend and the divisor are already arranged in the required manner.

Step 2: Perform Long Division

Now perform long division as follows: First, divide the highest degree term of the dividend \(2 x^{4}\) by the highest degree term of the divisor \(x^{2}\). We get \(2x^2\) as the first term of the quotient. Multiply the divisor \(x^{2}+5\) by the part of quotient \(2x^2\) to get \(2x^4 + 10x^2\), then subtract this from the dividend \(2 x^{4}+x^{2}-11\) to get \(-9x^2 -11\). Next, divide the highest degree term of the new dividend \(-9x^2\) by the highest degree term of the divisor \(x^2\) to get the next term of the quotient as \(-9\). Multiply the divisor with \(-9\) to get \(-9x^2 - 45\) and subtract this from the new dividend to get the remainder as 34.

Step 3: Write Down the Solution

The solution could be written down as the quotient \(2x^2 - 9\) and reminder 34.

Key Concepts

Long Division

Long division is a method often used to divide polynomials. Just like with numbers, the process involves dividing, multiplying, and subtracting to find the quotient and remainder. In polynomial long division, the goal is to divide a polynomial (the dividend) by another polynomial of lower or equal degree (the divisor).

This process begins by ensuring both polynomials are arranged in descending powers of the variable, meaning the terms are listed from the highest degree to the lowest.

• Identify the leading term (the term with the highest power) in the dividend and the divisor.
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this newly found term and subtract the result from the dividend.
• Repeat this process until no terms remain, or the degree of the remaining polynomial is less than the degree of the divisor.

Understanding long division in algebra is crucial as it lays the groundwork for more complex polynomial operations and theorems.

Polynomial Quotient

The polynomial quotient is the result obtained from dividing one polynomial by another. When using polynomial long division, the quotient is formed by all the terms found through the division process of the leading terms.

For instance, in the division of \( (2x^4 + x^2 - 11) \) by \( (x^2 + 5) \), the quotient happened to be \( 2x^2 - 9 \). It's essential to note that:

• The degree of the quotient is determined by subtracting the degree of the divisor from the degree of the dividend.
• The quotient can be a polynomial of one or several terms depending on the degrees and coefficients involved.

Gaining a clear understanding of how the quotient is constructed during division helps when verifying results or simplifying expressions in polynomial operations.

Remainder Theorem

The Remainder Theorem is an essential concept that connects division and evaluation in polynomials. It states that when a polynomial \( f(x) \) is divided by \( (x-a) \), the remainder of this division is \( f(a) \).

While the Remainder Theorem primarily applies to linear divisors, it illustrates a fundamental property of polynomial division. In the case of dividing \( (2x^4 + x^2 - 11) \) by \( (x^2 + 5) \), the remainder was 34, as found through the long division process. It's important to remember that:

• The presence of a non-zero remainder indicates that the divisor is not a factor of the dividend.
• In polynomial division where the divisor isn't a linear factor, we find the remainder directly, rather than by calculating \( f(a) \).

Understanding the Remainder Theorem's implications aids in mastering polynomial division and identifying zeros of polynomials.

Polynomial Operations

Polynomial operations encompass addition, subtraction, multiplication, and division of polynomials. Each operation follows algebraic rules, similar to those for integers, but focuses on managing terms with different degrees.

In division, particularly, understanding both the method and the final result is vital for simplifying complex expressions and solving equations. Other key operations involve:

• Addition and subtraction: Combine like terms, which have the same power of the variable.
• Multiplication: Use distributive property and apply the rule of exponents, \( x^m \cdot x^n = x^{m+n} \).
• Division: As mentioned, use long division or synthetic division, particularly for linear divisors.

Mastery of these polynomial operations is fundamental for progressing in algebra and calculus, allowing students to break down complex equations into manageable parts.