How To Know If A Polynomial Is Divisible

Partition of Polynomials: Definition, Method, and Examples

Division of Polynomials: Polynomials are algebraic expressions consisting of variables and constants such that the exponent on the variables is a whole number. We can perform arithmetic operations such as addition, subtraction, multiplication, and division with polynomials. While using segmentation performance we separate polynomial from a polynomial. It is to be noted that the division of 2 polynomial may or may not result in a polynomial.

Polynomial is derived from the Greek word. Poly means many and nomial means terms, so together, we can phone call a polynomial every bit many terms. So a polynomial has one or more than one term. The division of polynomials follows the same rules that we employ to follow in the sectionalisation of integers. This article details polynomials, their caste, types, and how to bear out the sectionalization of polynomials.

Polynomial Definition

Polynomials are algebraic expressions consisting of variables and constants with a whole number exponents of the variables. A polynomial looks like this:

A polynomial \(p(10)\) in one variable \(ten\) is an algebraic expression in \(x\) of the course
\(p(x) = {a_n}{x^due north} + {a_{north – 1}}{x^{n – i}} +  \ldots  + {a_2}{x^2} + {a_1}x + {a_0}\)
where \({a_0},\,{a_1},\,{a_2},\, \ldots ,\,{a_n}\) are constants and \({a_n} \ne 0\).
\({a_0},\,{a_1},\,{a_2},\, \ldots ,\,{a_n}\) are respectively the coefficients of \({ten^0},{x^one},{x^2},…,{x^n}\), and \(northward\) is called the degree of the polynomial, which should be a whole number.

Each of \({a_n}{x^n},\,{a_{north – 1}}{10^{n – ane}},\, \ldots ,\,{a_0}\), with \({a_n} \ne 0\), is called a term of the polynomial \(p(x)\). It is also important to note that a polynomial tin can't accept fractional or negative exponents.
Examples of polynomials are \(3{y^two} + 2x + 5,\,{ten^iii} + ii{x^ii} – 9x – four,\,x{x^three} + 5x + y,\,four{x^ii} – 5x + seven\) etc.

Types of Polynomials

Polynomials are classified based on the number of terms and their degree. Let's see both types of nomenclature.

Based on the Number of Terms

Based on the number of terms, we tin can allocate polynomials into three types, monomial, binomial, and trinomial.

1. Monomial: A polynomial having only one term is chosen a monomial.
Examples of monomials are \(5,\,2x,\,iii{a^ii},\,4xy\), etc.

two. Binomial: A polynomial having two terms separated by either the addition \(( + )\) or subtraction sign \(( – )\) is called a binomial.
Examples of binomial expressions are \(2x + 3,\,3x – ane,\,2x + 5y,\,6x – 3y\), etc.

3. Trinomial: A polynomial having exactly 3 terms is chosen trinomial.
Examples of trinomials are \(4{x^2} + 9x + 7,\,12pq + 4{x^2} – ten,\,3x + 5{x^2} – 6{10^3}\) etc.

Based on the Caste of a Polynomial

Based on the degree, we tin allocate the polynomials as zero or constant polynomial, linear polynomial, quadratic polynomial, cubic polynomial, quartic or quaternary-degree polynomial, and so on.

1. Constant or Nada Polynomial: A polynomial whose power of the variable is zero is known as a constant or cypher polynomial. When the ability of the variable is zero, its value is nothing only \(1\) as \({10^0} = 1\). The nothing polynomials will have terms that are constants like \(2,\,5,\,10,\,101\) etc.
Instance: \(3{x^0} = 3 \times 1 = 3\).

ii. Linear Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(i\) is a linear polynomial.
Example: \(10 – 1,\,y + one,\,a + 4\), etc.

3. Quadratic Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(ii\) is a quadratic polynomial.
Example: \({10^two} + x,\,{y^2} + 1,\,{a^2} + eight\), etc.

4. Cubic Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(3\) is a cubic polynomial.
Instance: \({y^3} + 8,\,{ten^iii} – 27,\,5 + {a^3},\,{ten^iii} + {x^2} – x + ii\) etc.

v. Quartic Polynomial or fourth-degree polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(4\) is known as a quartic polynomial or 4th-degree polynomial.
Instance: \({10^4} + {x^3} – {x^ii} + x + ane,\,{y^4} – {y^2} + 1\), etc.

Sectionalisation of Polynomials Methods

In algebra, we can perform the segmentation of algebraic expressions in three means:

1. Sectionalisation of a monomial past another monomial
2. Division of a polynomial by a monomial
iii. Division of a polynomial by another polynomial

Nosotros all are familiar with the bones division algorithm formula of dividing numbers.
\({\rm{Dividend = (Divisor \times Quotient) + Remainder}}\)

A long division of polynomials is a method for dividing a polynomial by another polynomial of the same or a lower degree. In the long division of polynomials, numerator and denominator are polynomials, as shown below.

The long division of polynomials consists of a divisor, a caliber, a dividend, and a remainder.

Steps for Long Sectionalisation of Polynomials

The following are the steps required for dividing by a polynomial containing more than one term (Long partition method):

Pace 1: Write the polynomial in descending lodge(bundled from the largest caste to the smallest degree). If any term is missing, then use a aught in place of the missing term.
For example: Let us divide the polynomial \(a(x) = 6{x^4} + 3x – 9{x^2} + 6\) past the quadratic polynomial \(b(x) = {x^2} – ii\) by using the long partition method.

Outset, arrange the given polynomial in the descending order of the power of the variable.

Step 2: Add together the missing terms with zero every bit the coefficient. Dissever the polynomial \(a(x)\) by \(b(x)\) using the aforementioned method that we utilise to carve up the numbers.
\(a(x):6{10^4} + 0{x^3} – 9{10^2} + 3x + vi = {ten^2} – ii\)
\(b(ten):{x^ii} + 0x – ii\)

Pace iii: Separate the first term of the dividend by the beginning term of the divisor.
Now, carve up \(half-dozen{x^4}\) past \({x^2}\) nosotros become the \({{\rm{ane}}^{{\rm{st}}}}\) term of the quotient is  \(half dozen{x^ii}\).

Pace 4: Multiply the quotient obtained in the previous stride by the divisor.
That is, multiply the divisor by \(6{10^2}\).
\(6{x^ii} \times \left( {{ten^2} + 0x – two} \right)\)
Then, we will get the product under the dividend equally
\(half dozen{10^4} + 0{10^3} – 12{ten^2}\)

Step 5: Subtract the obtained product from the dividend and then write down the next terms to get the new dividend.

Pace 5: Repeat the process in Steps \(2,\,3\), and \(four\) until nosotros get remainder or no more terms to bring downwardly.
Dissever \(3{x^2}\) by \({x^2}\), we become the \({{\rm{2}}^{{\rm{nd}}}}\) term of the quotient equals \(3\).

The power of the remaining dividend \(3x\) is \(1\). It is less than the power of the divisor which is \(2\). Then, we get the required non-nil remainder.

Notation: Since the balance is not-zero, so nosotros tin say that \({10^ii} – 2\) is not a factor of \(half dozen{x^iv} – 9{10^2} + 3x + 6\).
If the remainder is zero then we can say that \({x^2} – 2\) is a factor of \(6{x^4} – nine{x^two} + 3x + half-dozen\).

Solved Examples

Q.1. Divide the polynomial \(ii{x^2} + 3x + 1\) past \(x + 2\).
Ans:

And then, here the quotient is \(2x – 1\) and the remainder is \(3\).
Also, \((2x – i)(x + 2) + 3 = 2{ten^ii} + 3x – two + 3 = 2{x^2} + 3x + 1\)
i.e., \(2{x^2} + 3x + 1 = (x + ii)(2x – one) + 3\)

Therefore, \({\rm{Dividend = Divisor \times Quotient + Residue}}\)

Q.2. Detect the remainder obtained when \({x^iv} + {x^iii} – 2{x^ii} + x + 1\) is divided by \(x – 1\).
Answer: It is given that, dividend \( = {x^4} + {ten^3} – 2{x^two} + ten + 1\) and divisor \( = x – 1\).
Let us divide the polynomial.

So, here the quotient is \({10^3} + two{x^2} + 1\) and the residuum is \(2\).
Hence, \(2\) is the residuum when \({x^4} + {x^3} – 2{x^2} + x + 1\) is divided by \(x – i\).

Q.three. Find all the zeroes of \(2{ten^four} – 3{x^3} – 3{x^2} + 6x – ii\), if the two of its zeroes are \(\sqrt 2 \) and \( – \sqrt 2 \).
Ans: Since two zeroes are \(\sqrt two \)  and \(\sqrt two ,\,(x – \sqrt 2 )(ten + \sqrt ii ) = {x^ii} – 2\) is a gene of the given polynomial. Now, we divide the given polynomial past \({x^two} – two\).

So, \(2{x^4} – 3{x^3} – 3{x^2} + 6x – 2 = \left( {{ten^2} – 2} \right)\left( {ii{x^2} – 3x + ane} \correct)\) Now, by splitting \( – 3x\), nosotros factorise \(ii{x^2} – 3x + 1\) equally \((2x – 1)(10 – ane)\).
And then, its zeroes are \(ten = \frac{ane}{2}\) and \(10 – 1\).

Hence, the required zeroes of the given polynomial are \(\sqrt ii ,\, – \sqrt 2 ,\,\frac{1}{two}\), and \(1\).

Q.4. Separate the polynomial \(3{x^2} – {x^3} – 3x + 5\) by the trinomial \(x – 1 – {x^2}\), and verify the sectionalisation algorithm.
Ans: Notation that the given polynomials are not in standard form. To carry out partitioning, nosotros commencement write both the dividend and divisor in decreasing orders of their degrees.
And so, dividend \( =  – {ten^3} + iii{10^ii} – 3x + 5\) and divisor \( =  – {ten^2} + x – 1\).

Since degree \({\rm{(iii) = 0 < 2 = }}\) degree \(\left( { – {x^two} + x – i} \right)\), so, quotient \( = ten – ii\), remainder \(=ii\).
Now, the formula for the sectionalisation algorithm is
\({\rm{ Divisor }} \times {\rm{ Quotient }} + {\rm{ Remainder }}\)
\( = \left( { – {x^2} + ten – 1} \right)(x – 2) + 3\)
\( =  – {10^3} + {x^2} – x + two{x^two} – 2x + two + iii\)
\( =  – {x^3} + 3{x^two} – 3x + 5 = {\rm{Dividend }}\)
Hence, the sectionalisation algorithm is verified.

Question 5: Divide the polynomial \(12 – 14{a^two} – 13a\) by the binomial \(three + 2a\).
Answer: Note that the given polynomials are non in standard form. To carry out division, nosotros outset write both the dividend and divisor in decreasing orders of their degrees.
So, dividend \( =  – 14{a^two} – 13a + 12\)  and divisor \( = 2a + iii\).

Summary

In this article, we have learnt about polynomials, how information technology looks like, types of polynomials based on the number of terms similar monomials, binomials and trinomials. And based on the degree, polynomials are farther classified into zero-degree polynomial or constant polynomial, linear polynomial, quadratic polynomial, cubic polynomial, quartic polynomial, etc.

Then we have discussed the method of division of polynomials by using the long division method, steps for division of polynomials, and solved examples.

FAQs – Segmentation of Polynomials

Q.1. How do you divide polynomials with two variables?
Ans : We can dissever the polynomials with two variables using the long sectionalisation method or the synthetic division method.

Q.2. What is the easiest way to dissever polynomials?
Ans: Past using the long division method, we can easily divide the polynomials to any degree.

Q.3. How is a polynomial division used in real life?
Ans: We employ polynomial division for several aspects of our daily lives. We use information technology in coding, engineering, business, designing, architecting, and other real-life areas. It is used to solve the problems related to solving the expressions involved in expanse and book.

Q.4. How exercise you divide polynomials?
Ans: Polynomials can be divided the same way as we divide numbers, either by factorising or by long division. The method you utilise depends upon how complex are the polynomial dividend and divisor.

Q.five. How do yous know if a polynomial completely divides another polynomial?
Ans: If the remainder is zero subsequently completing the division process, then information technology means that the polynomials are completely divisible. If a remainder is non-zero, then the polynomial is non divisible.

We hope this article on the partitioning of polynomials has provided significant value to your knowledge. If yous have any queries or suggestions, please write them downwardly in the comment section below. Nosotros will beloved to hear from you. Embibe wishes y'all all the best of luck!

Source: https://www.embibe.com/exams/division-of-polynomials/

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