Dividing Polynomials
Introduction
Dividing polynomials is a mathematical operation involving the division of one polynomial by another. Typically, the divisor polynomial has a lower degree compared to the dividend. This operation may or may not yield a polynomial as the result. In this article, we'll explore the polynomial division in depth.
What Is the Division of Polynomials?
Polynomials represent algebraic expressions comprising variables and coefficients, typically arranged in descending order of degrees. For instance, \(2x^2 + 3x - 8\) features three terms organized by their respective degrees, with the highest degree term first, followed by lower degree ones in ascending order of degree. Division of polynomials entails an algorithmic process used to compute a polynomial divided by a monomial, binomial or another polynomial. This division follows a similar structure to regular long division, with the divisor and dividend arranged accordingly. For example, when dividing \(2x^2 + 3x - 8\) by \(2x - 4\), it is represented as:
The polynomial above the division bar represents the numerator \(2x^2 + 3x - 8\), while the polynomial below the bar represents the denominator \(2x - 4\). This convention is illustrated in the following diagram, indicating that the numerator becomes the dividend and the denominator becomes the divisor.
Dividing Polynomial by Monomial
When dividing polynomials by monomials, there are two approaches to consider. One method involves separating the polynomial based on the addition and subtraction operators, addressing each part individually. Alternatively, the other method would be simplifying the expression through factorization and subsequently cancelling out the common factors across the numerator and the denominator. Let's explore both techniques in greater detail.
Splitting the Terms Method
In this method, we break down the polynomial into its individual terms, separating them by the operator '`+`' or '`-`'. Then, simplify each term individually.
Given the polynomial division `\frac{3x^3 + 2x^2 - 6x}{x}`, we can split it into individual terms:
`\frac{3x^3}{x} + \frac{2x^2}{x} - \frac{6x}{x}`
Now, let's simplify each term:
`\frac{3x^3}{x} = 3x^2`
`\frac{2x^2}{x} = 2x`
`\frac{-6x}{x} = -6`
Putting it all together, we get:
\( 3x^2 + 2x - 6 \)
Factorization Method
Certainly, when performing polynomial division, it's often necessary to factor the polynomial to identify common factors between the numerator and the denominator. For instance, consider dividing the polynomial \(6x^3 + 9x^2\) by \(3x\). Both the numerator and denominator share a common factor of \(3x\). Consequently, the expression can be expressed as `\frac{3x(2x^2 + 3x)}{3x}`. Canceling out the common term \(3x\), we're left with \(2x^2 + 3x\) as the quotient.
Dividing Polynomial by Binomial
There are various methods of dividing polynomials, some of those methods are:
- Long Division Method
- Synthetic Division Method
Dividing Polynomials - Long Division Method
Let's walk through the algorithm of dividing polynomials by binomials using a different example.
Example: Divide \( (6x^3 + 4x^2 - 2x - 60) ÷ (2x - 4) \).
Solution:
Here, \( (6x^3 + 4x^2 - 2x - 60) \) is the dividend, and \( (2x - 4) \) is the divisor, which is a binomial. Below, you'll find the division followed by the step-by-step process.
This technique involves arranging the polynomials in standard form, with terms ordered from the highest degree to the lowest, inserting `0`s as coefficients for any missing terms.
Step `1`: Divide the first term of the dividend (\(6x^3\)) by the first term of the divisor (\(2x\)), yielding \(3x^2\), which becomes the first term in the quotient.
Step `2`: Multiply the divisor by the quotient obtained (\(3x^2\)), giving \(6x^3 - 12x^2\), and place this product beneath the dividend, starting from the left.
Step `3`: Subtract the product from the dividend: \((6x^3 + 4x^2 - 2x - 60) - (6x^3 - 12x^2) = 16x^2 - 2x - 60\).
Step `4`: Now, we repeat the process with the new polynomial obtained after subtraction: \(16x^2 - 2x - 60\).
Step `5`: Divide the first term of this new polynomial (\(16x^2\)) by the first term of the divisor (\(2x\)), yielding \(8x\), which becomes the second term in the quotient.
Step `6`: Multiply the divisor by the new quotient term (\(8x\)), giving \(16x^2 - 32x\), and place this product beneath the polynomial, starting from the left.
Step `7`: Subtract the product from the polynomial: \((16x^2 - 2x - 60) - (16x^2 - 32x) = 30x - 60\).
Step `8`: Now, we have \(30x - 60\) left to consider.
Step `9`: Divide the first term of this remaining polynomial (\(30x\)) by the first term of the divisor (\(2x\)), yielding \(15\), which becomes the third term in the quotient.
Step `10`: Multiply the divisor by the new quotient term (\(15\)), giving \(30x - 60\), and place this product beneath the polynomial, starting from the left.
Step `11`: Subtract the product from the polynomial: \((30x - 60) - (30x - 60) = 0\).
Step `12`: Since the degree of \( 0 \) is less than that of \( (2x - 4) \), we can't continue dividing further.
Therefore, the quotient is \( 3x^2 + 8x + 15 \), and the remainder is \( 0 \).
Dividing Polynomials - Synthetic Division Method
Synthetic division offers a method to divide a polynomial by a linear binomial by solely focusing on the coefficients. Like the long division method, this technique involves arranging the polynomials in standard form, with terms ordered from the highest degree to the lowest, inserting `0`s as coefficients for any missing terms. For instance, \(5x^3 + 7\) should be written as \(5x^3 + 0x^2 + 0x + 7\).
However, synthetic division only works for linear divisors, unlike long division that works in all cases.
Below are the steps to perform polynomial division using synthetic division:
Example: Divide \(2x^3 - 5x^2 + 3x - 7\) by \(x - 2\) using synthetic division.
Solution:
Below are the steps illustrated in the image above:
Step `1`: Write the divisor in the form of \(x - k\), and identify \(k\), which is \(2\) in this case.
Step `2`: Set up the division by writing the coefficients of the dividend on the right side and \(k\) (which is \(2\)) on the left side. If there are any missing terms in the dividend, represent them with \(0\)'s.
Step `3`: Bring down the coefficient of the highest degree term of the dividend, which is \(2\) (the coefficient of \(x^3\)).
Step `4`: Multiply \(k\) which is (\(2\)) with the leading coefficient (\(2\)) and write the product below the second coefficient from the left side of the dividend. So, we get \(2 \times 2 = 4\), which we'll write below \(-5\).
Step `5`: Add the numbers written in the second column. Here, by adding, we get \(-5 + 4 = -1\).
Step `6`: Repeat the process of multiplying \(k\) with the number obtained in Step `5` (which is \(-1\)) and write the product in the next column to the right.
Step `7`: Finally, write the final answer, which will be one degree less than the dividend. Since the highest degree term in the dividend is \(x^3\), the quotient will have the highest degree term as \(x^2\). Therefore, the answer obtained is `2x^2 - x + 1 - \frac{5}{x - 2}`. Please note that \(-5\) is the remainder here.
Solved Examples
Example `1`: What is the quotient when \(3x^2 - 7x + 5\) is divided by \(x - 2\)?
Solution:
So, `\frac {3x^2 - 7x + 5}{x-2} = 3x - 1 + \frac{3}{x - 2}`.
The quotient when \(3x^2 - 7x + 5\) is divided by \(x - 2\) is \(3x - 1\).
Example `2`: Divide \(2x^3 - 5x^2 + 3x + 7\) by \(x - 1\) using synthetic division.
Solution:
So, `\frac {2x^3 - 5x^2 + 3x + 7}{x-1} = 2x^2 - 3x + \frac{7}{x - 1}`.
Example `3`: Divide \(9x^5 - 6x^4 + 3x^3 - 2x^2 + x - 5\) by \(x - 1\).
Solution:
So, `\frac {9x^5 - 6x^4 + 3x^3 - 2x^2 + x - 5}{x-1} = 9x^4 + 3x^3 + 6x^2 + 4x +5`.
Practice Problems
Q`1`. What is the result when \(3x^2 + 2x - 5\) is divided by \(x - 3\)?
- `2x + 9 + \frac{36}{x - 4}`
- `5x + 8 + \frac{20}{x + 1}`
- `3x + 11 + \frac{28}{x - 3}`
- `7x + 12 + \frac{30}{x - 5}`
Answer: c
Q`2`. What is the result of \(4x^3 - 6x^2 + 2x + 1\) divided by \(2x + 1\)?
- `2x^2 - 4x + 3 - \frac{2}{2x + 1}`
- `2x^2 + 4x + 3 - \frac{3}{2x + 1}`
- `x^2 - 4x + 3 - \frac{1}{x + 2}`
- `2x^2 - 7x + 6 - \frac{4}{2x + 2}`
Answer: a
Q`3`. When \(6x^3 + 3x^2 - 8x - 15\) is divided by \(2x - 3\), the quotient is:
- \(3x^2 - 2x + 4\)
- \(3x^2 + 6x - 5\)
- \(3x^2 - 2x - 4\)
- \(3x^2 + 6x + 5\)
Answer: d
Q`4`. Find the quotient when \(9x^3 - 6x^2 + 4x +8\) is divided by \(3x + 2\).
- \(3x^2 - 2x + 6\)
- \(3x^2 - 4x + 4\)
- \(3x^2 + 2x - 6\)
- \(3x^2 + 2x + 6\)
Answer: b
Q`5`. Find the quotient when \(5x^3 + 2x^2 - 4x - 8\) is divided by \(x + 2\).
- `5x^2 - 8x + 12 - \frac{32}{x + 2}`
- `5x^2 + 8x + 12 - \frac{32}{x + 2}`
- `5x^2 - 10x + 12 - \frac{32}{x + 2}`
- `5x^2 + 10x + 12 - \frac{32}{x + 2}`
Answer: a
Frequently Asked Questions
Q`1`. What is polynomial division?
Answer: Polynomial division is the process of dividing one polynomial expression by another, yielding a quotient and/or a remainder.
Q`2`. How do you divide polynomials?
Answer: Use long division or synthetic division, ensuring the divisor's degree is less than or equal to the dividend's degree.
Q`3`. What's the importance of polynomial division?
Answer: It helps simplify complex expressions, find roots of polynomials, and solve various mathematical problems.
Q`4`. What's the difference between long division and synthetic division?
Answer: Long division is more versatile and applicable to any polynomial division, while synthetic division works specifically for linear divisors.
Q`5`. When is polynomial division used in real life?
Answer: It's utilized in engineering for circuit analysis, physics for modeling physical systems, and economics for optimizing functions.