5 4 Practice Dividing Polynomials

Dividing Polynomials

Dividing polynomials is an arithmetics operation where we divide a polynomial past another polynomial, more often than not with a lesser degree equally compared to the dividend. The division of two polynomials may or may non result in a polynomial. Let's learn about dividing polynomials in this article in detail.

one.	What is Dividing Polynomials?
two.	Dividing Polynomials by Monomials
three.	Dividing Polynomials by Binomials
4.	Dividing Polynomials Using Constructed Partition
5.	FAQs on Dividing Polynomials

What is Dividing Polynomials?

Polynomials are algebraic expressions that consist of variables and coefficients. It is written in the following format: 5x² + 6x - 17. This polynomial has three terms that are arranged co-ordinate to their degree. The term with the highest degree is placed start, followed past the lower ones. Dividing polynomials is an algorithm to solve a rational number that represents a polynomial divided past a monomial or another polynomial. The divisor and the dividend are placed exactly the same way every bit we do for regular division. For example, if we need to carve up 5x² + 7x + 25 by 6x - 25, nosotros write information technology in this way:

\[\dfrac{(5 x ^ii+7 x+25)}{(6 10 -25)}\]

The polynomial written on top of the bar is the numerator ( 5xⁱⁱ + 7x + 25), while the polynomial written below the bar is the denominator (6x - 25). This tin can be understood past the post-obit figure which shows that the numerator becomes the dividend and the denominator becomes the divisor.

dividing polynomials

Dividing Polynomials by Monomials

While dividing polynomials by monomials, the division can be done in ii ways. One is by simply separating the '+' and '-' operator signs. That means, we break the polynomial from the operating sign, and solve each part separately. Some other method is to do the simple factorization and further simplifying. Allow us take a expect at both methods in detail.

Splitting the Terms Method

Split the terms of the polynomial separated by the operator ( '+' or '-' ) between them and simplify each term. For instance, (4xⁱⁱ - 6x) ÷ (2x) can be solved as shown here. We first take common terms from the numerators and denominators of both the terms, we become, [(4x²) / (2x)] - [(6x) / (2x)]. Canceling the common term 2x from the numerator and the denominator, we get 2x - 3.

Factorization Method

When you divide polynomials you may have to factor the polynomial to detect a common gene between the numerator and the denominator. For instance: Dissever the following polynomial: (2x² + 4x) ÷ 2x. Both the numerator and denominator have a common factor of 2x. Thus, the expression tin be written as 2x(x + two) / 2x. Canceling out the common term 2x, we get 10+ii as the answer.

Dividing Polynomials past Binomials

For dividing polynomials by binomials or any other type of polynomials, the most common and general method is the long division method. When there are no common factors betwixt the numerator and the denominator, or if y'all can't find the factors, you can use the long sectionalisation process to simplify the expression.

Dividing Polynomials Using Long Division

Let us go through the algorithm of dividing polynomials by binomials using an example: Carve up: (4x² - 5x - 21) ÷ (ten - three). Here, (4x² - 5x - 21) is the dividend, and (x - iii) is the divisor which is a binomial. Observe the division shown below, followed by the steps.

dividing polynomial by binomial

Step 1. Divide the kickoff term of the dividend (4x²) by the showtime term of the divisor (ten), and put that as the kickoff term in the quotient (4x).

Step 2. Multiply the divisor by that respond, place the product (4x² - 12x) beneath the dividend.

Stride 3. Subtract to create a new polynomial (7x - 21).

Stride 4. Echo the aforementioned process with the new polynomial obtained after subtraction.

And so, when we are dividing a polynomial (4x² - 5x - 21) with a binomial (ten - 3), the quotient is 4x+7 and the remainder is 0.

Dividing Polynomials Using Constructed Division

Synthetic partitioning is a technique to divide a polynomial with a linear binomial past only considering the values of the coefficients. In this method, we first write the polynomials in the standard class from the highest degree term to the lowest degree terms. While writing in descending powers, use 0's as the coefficients of the missing terms. For example, x³+3 has to be written every bit x³ + 0x² + 0x + iii. Follow the steps given beneath for dividing polynomials using the synthetic division method:

Let u.s.a. separate xⁱⁱ + three by ten - 4.

Step 1: Write the divisor in the class of x - k and write chiliad on the left side of the division. Here, the divisor is ten-4, so the value of one thousand is 4.

Step 2: Set up the division by writing the coefficients of the dividend on the right and chiliad on the left. [Notation: Use 0's for the missing terms in the dividend]

Step 3: Now, bring down the coefficient of the highest degree term of the dividend as it is. Here, the leading coefficient is 1 (coefficient of 10ⁱⁱ).

Step iv: Multiply thousand with that leading coefficient and write the product below the second coefficient from the left side of the dividend. Then, we get, 4×i=4 that we volition write below 0.

Step 5: Add the numbers written in the 2d cavalcade. Here, past adding we go 0+4=4.

Pace 6: Repeat the same process of multiplication of 1000 with the number obtained in step 5 and write the production in the next cavalcade to the right.

Step 7: At last, we volition write the final answer which will be i caste less than the dividend. And so, here, in our dividend, the highest degree term is ten², therefore, in the quotient, the highest degree term will be x. Therefore, the answer obtained is x+4+(19/x-4).

dividing polynomials using synthetic division

Topics Related to Dividing Polynomials

Check these articles to know more than virtually the concept of dividing polynomials and its related topics.

Long sectionalization of polynomials
Synthetic division of polynomial
Segmentation Algorithm for Polynomials
Dividing 2 Polynomials
Partition of Polynomial by Linear Factor

Dividing Polynomials Examples

Example 1: Alex is stuck on a problem while working on dividing polynomials. Tin you help him to obtain the caliber: (x⁴ - 10x³ + 27x² - 46x + 28) ÷ (ten - 7).

Solution:

Therefore, the quotient is x³ - 3x²+ 6x - 4.
Case 2: Stacy needs aid in finding the remainder while dividing polynomials. Can yous help her solve this?

(4x³ + 5x² + 5x + viii) ÷ (4x + 1)

Solution: Using the long division method of dividing polynomials, we become,

Therefore, the residual is vii.

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FAQs on Dividing Polynomials

What is Dividing Polynomials?

Dividing polynomials is ane of the arithmetics operations performed on two given polynomials. In this, the dividend is by and large of a higher degree and the divisor is a lower caste polynomial.

Why is Dividing Polynomials Important?

Dividing polynomials is important because it provides an algorithm to solve a rational number that represents a polynomial divided by a monomial or another polynomial.

What is the Easiest Way for Dividing Polynomials?

The easiest mode to separate polynomials is by using the long division method. Notwithstanding, in the case of the division of polynomials by a monomial, it can be direct solved by splitting the terms or by factorization.

What are the Two Methods of Dividing Polynomials?

The two methods to split up polynomials are given below:

Synthetic division method
Long division method

What is Polynomial Division Used for in Existent-Life?

We use polynomial division for various aspects of our day-to-mean solar day lives. We demand it for coding, applied science, designing, architecting, and diverse other existent-life areas.

What Method of Dividing Polynomials by a Monomial is All-time?

In the case of the division of polynomials by a monomial, it tin be direct solved by splitting the terms or by factorization. Nosotros can divide each term of the dividend with the given monomial and find the consequence.