use synthetic division to solve x^3-x^2-17x-15 what is the quotient

In Mathematics, there are two different methods to divide the polynomials. One is the long division method. Another i is the synthetic division method. Among these two methods, the shortcut method to divide polynomials is the constructed division method. It is also called the polynomial partitioning method of a special instance when it is dividing by the linear factor.  It replaces the long division method.  In sure situations, you can find this method easier. In this article, we volition hash out what the synthetic partition method is, how to perform this method, steps with more than solved examples.

Table of Contents:

• Definition
• How to Perform Synthetic Partitioning
• Steps
• Advantages and Disadvantages
• Examples
• Practice Questions
• FAQs

Synthetic Division of Polynomials

The Synthetic sectionalization is a shortcut way of polynomial division, specially if nosotros need to divide it by a linear factor. Information technology is generally used to notice out the zeroes or roots of polynomials and not for the division of factors. Thus, the formal definition of synthetic segmentation is given equally:

"Synthetic sectionalisation tin can be divers equally a simplified mode of dividing a polynomial with another polynomial equation of caste 1 and is generally used to detect the zeroes of polynomials"

This division method is performed manually with less effort of adding than the long partition method. Unremarkably, a binomial term is used as a divisor in this method, such as ten – b.

If we split up a polynomial P(x) by a linear cistron (x-a), which of the polynomial of the degree 1, Q(x) is quotient polynomial and R is the residual, which is a constant term. We apply the synthetic division method in the context of the evaluation of the polynomial using the remainder theorem, wherein nosotros evaluate the polynomial P(x) at "a" while dividing the polynomial P(x) past the linear gene. (i.e) P(x)/(x-a).

Mathematically, it can be represented as follows:

P(x)/Q(x) = P(10)/(x-a) = Quotient + [Residuum/(x-a)]

(i.e)

P(ten)/(x-a) = Q(10) +[R/(x-a)]

Hence, nosotros can utilise the constructed division method to discover the remainder rapidly, if "a" is the gene of the polynomial.

In the constructed division method, nosotros use only the numbers for adding and this method avoids the usage of the variables.

Note:

• We can perform the synthetic division method, only if the divisor is a linear gene.
• In the constructed division method, we will perform multiplication and addition, in the place of sectionalisation and subtraction, which is used in the long division method.

How to Perform a Synthetic Division?

If nosotros want to split polynomials using synthetic division, y'all should be dividing information technology by a linear expression and the first number or the leading coefficient should be a 1. This division by linear denominator is also called division through Ruffini'south dominion(newspaper-and-pencil computation).

The requirements to perform the synthetic procedure method is given below:

• The divisor of the given polynomial should be of caste 1. It means that the exponent of the given variable should be 1. Such kind of divisor is considered as the linear factor.
• The coefficient of the divisor variable (say x) should be also equal to 1.

The process of the synthetic division volition get messed upwardly if the divisor of the leading coefficient is other than 1. In case if the leading coefficient of the divisor is other than ane while performing the synthetic division method, solve the problem advisedly.

The basic Mantra to perform the constructed sectionalization procedure is"

"Bring down, Multiply and add, multiply and add, Multiply and add, …."

For example, we can use the constructed division method to split up a polynomial of 2 degrees by x + a or x – a, but y'all cannot utilise this method to divide past x² + 3 or 5x² – x + vii.

If the leading coefficient is not 1, then we need to divide by the leading coefficient to turn the leading coefficient into 1. For example, 4x – ane would get 10 – ¼ and 4x+nine would go x + 9/four. If the synthetic partitioning is not working, and so we need to apply long division.

• Multiplying Polynomials
• Factoring Polynomials
• Polynomial Partitioning
• Euclid Division Lemma
• Remainder Theorem And Polynomials

Steps for Polynomial Synthetic Division Method

Post-obit are the steps required for Synthetic Division of a Polynomial:

Step 1	To set upwards the problem, we need to set the denominator = zero, to observe the number to put in the division box. Then, the numerator is written in descending order and if whatever terms are missing we need to use a nada to fill in the missing term. At terminal, list only the coefficient in the division problem.
Step two	Now, when the trouble is set up perfectly, bring the offset number or the leading coefficient straight down.
Step iii	So, put the consequence in the next column past multiplying the number in the division box with the number you brought down.
Pace 4	Write the result at the lesser of the row by calculation the two numbers together
Step 5	Until y'all achieve the end of the problem, repeat steps iii and 4.
Footstep 6	Write the final answer. The numbers in the lesser row with the last number being the remainder and the remainder which is written as a fraction makes the final answer. The variables shall showtime with one ability less than the real denominator and go down ane with each term.

Advantages and Disadvantages of Synthetic Division Method

The advantages of using the synthetic division method are:

• It requires only a few adding steps
• The calculation can be performed without variables
• Dissimilar the polynomial long division method, this method is a less fault-decumbent method

The only disadvantage of the synthetic division method is that this method is but applicable if the divisor of the polynomial expression is a linear cistron.

Synthetic Division Examples

Example 1:

Dissever :

\(\brainstorm{array}{l}\frac{2x^{3} – 5x^{two} + 3x + 7}{x-ii}\stop{array} \)

Solution:

Following the steps equally per explained above, to split the polynomials given. Thus, nosotros can get;

Synthetic Sectionalisation Example ane

Example ii:

Split up :

\(\begin{array}{l}\frac{2x^{iii} + 5x^{2} + nine}{x+3}\end{array} \)

Solution:

As per the given question; we have two polynomials in numerator and denominator. The denominator consists of a linear equation, and then nosotros can hands apply the constructed division method here.

Follow the footstep by step method every bit given beneath:

Instance 3:

Split up :

\(\begin{array}{l}\frac{3x^{3} + 5x – one}{ten+one}\stop{array} \)

Solution:

Following the same steps as per previous examples.

Synthetic Sectionalization Case 3

Instance 4:

Divide :

\(\brainstorm{assortment}{l}\frac{4x^{3} – 8x^{2} -10 +v}{2x – 1}\end{array} \)

Solution:

As we know, the step to solve the given equation by synthetic partitioning method, we can write;

Example five:

Divide :

\(\begin{assortment}{l}\frac{x^{3} – 5x^{2} +3x + 7}{x – three}\end{array} \)

Solution:

Solving the given expression, past step by pace method, we go;

Constructed Sectionalisation of polynomials Practice Questions

Solve the post-obit problems:

1. Discover the caliber and residual of the polynomial 2x ^three -7x ^two +0x+xi, when it is divided by a linear factor x-3.
2. Solve the following polynomial equation and find its quotient and residuum.(9a²-39a-30)/(a-5)
3. Notice Q(x) and R for the polynomial, P(x)=m³-3m+4 divided by the linear factor m-1.

Oft Asked Questions on Synthetic Division

What is meant by constructed segmentation?

The synthetic division method is a special method of dividing polynomials. This method is a special case of dividing a polynomial expression by a linear gene, in which the leading coefficient should exist equal to 1.

What are the requirements of the synthetic division method?

The requirements of the synthetic partitioning method are:
The divisor of the polynomial expression must have a caste of one (linear factor)
The leading coefficient of the variable in the divisor should be equal to i.

What is the Chief Use of Synthetic Division?

Synthetic division is mainly used to find the zeroes of roots of polynomials.

When Can You lot Use Synthetic Partition?

Constructed division is used when a polynomial is to be divided by a linear expression and the leading coefficient (starting time number) must be a ane. For example, any polynomial equation of whatsoever degree can be divided by x + 1 but not past x²+1

Why is Synthetic Division Of import?

Synthetic division is useful to divide polynomials in an easy and simple way as information technology breaks down complex equations into smaller and easier equations.

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