polynomial long division what to do with remainder

April 11, 2022 Post a Comment

The polynomial division involves the segmentation of one polynomial by another. The segmentation of polynomials can be betwixt 2 monomials, a polynomial and a monomial or between two polynomials. Before learning how to carve up polynomials, let's have a brief introduction to the definition of polynomial and its related terms.

Polynomial:

A polynomial is an algebraic expression of the type a_northwardxⁿ+ a_{due north−i}xⁿ⁻¹+…………………a₂x²+ a_ix + a₀, where "n" is either 0 or positive variables and real coefficients.

In this expression, a_{due north}, a_n−1…..a₁, a₀ are coefficients of the terms of the polynomial.

The highest power of x in the to a higher place expression, i.e. due north is known as the caste of the polynomial.

If p(ten) represents a polynomial and ten = thou such that p(thousand) = 0 and so yard is the root of the given polynomial.

Tabular array of Contents:

Division of Polynomials
Types of Polynomial Division
Monomial by Monomial
Polynomial past Monomial
Polynomial by Binomial
Polynomial by another Polynomial
Division algorithm
Practise problems

Example: Given a polynomial equation, p(x)=x²–x–2. Notice the zeroes of the equation.

Solution:

Given Polynomial, p(x)=x²–x–two

Zeroes of the equation is given past:

ten²–2x+x–2=0

x(ten−2)+one(x–2)

(x+1)(10−2)=0

⇒ x=−i

Or, 10=2

Thus, -i and two are zeroes of the given polynomial.

It is to exist noted that the highest power(caste) of the polynomial gives the maximum number of zeroes of the polynomial.

Segmentation of Polynomial

The sectionalisation is the process of splitting a quantity into equal amounts. In terms of mathematics, the process of repeated subtraction or the reverse operation of multiplication is termed as division. For example, when xx is divided by 4 we get 5 as the result since four is subtracted 5 times from 20.

The 4 bones operations viz. addition, subtraction, multiplication and division tin too be performed on algebraic expressions. Let us empathise the procedure and different methods of dividing polynomials and algebraic expressions.

Types of Polynomial Division

For dividing polynomials, by and large, three cases tin arise:

Partitioning of a monomial by another monomial
Segmentation of a polynomial by monomial
Division of a polynomial by binomial
Division of a polynomial by some other polynomial

Allow us discuss all these cases ane by one:

Division of a monomial by another monomial

Consider the algebraic expression 40x² is to be divided by 10x then

40x²/10x = (two×2×5×2×x×10)/(2×5×x)

Here, 2, 5 and x are mutual in both the numerator and the denominator.

Hence, 40x²/10x = 4x

Segmentation of a polynomial past monomial

The second example is when a polynomial is to be divided past a monomial. For dividing polynomials, each term of the polynomial is separately divided past the monomial (as described in a higher place) and the quotient of each division is added to go the result. Consider the post-obit instance:

Example: Split up 24x³ – 12xy + 9x by 3x.
Solution: The given expression 24x^three – 12xy + 9x has three terms viz. 24x³, – 12xy and 9x. For dividing the polynomial with a monomial, each term is separately divided as shown below:(24x³–12xy+9x)/3x = (24x^three/3x)–(12xy/3x)+(9x/3x) = 8x²–4y+3

Division of a Polynomial by Binomial

As we know, binomial is an expression with two terms. Dividing a polynomial by binomial tin be done easily. Hither, first nosotros need to write the given polynomial in standard form. Now, using the long sectionalisation method, nosotros can divide the polynomial equally given below.

Example: Divide 3x^three – 8x + 5 past x – 1.

Solution:

The Dividend is 3x³ – 8x + 5 and the divisor is x – 1.

Afterwards this, the leading term of the dividend is divided by the leading term of the divisor i.eastward. 3x³ ÷ ten =3xⁱⁱ.

This result is multiplied by the divisor i.e. 3xⁱⁱ(10 -i) = 3xⁱⁱⁱ -3xⁱⁱ and it is subtracted from the divisor.

Now again, this result is treated every bit a dividend and the same steps are repeated until the remainder becomes "0" or its degree becomes less than that of the divisor as shown beneath.

Polynomial Division

Division of Polynomial by Another Polynomial

For dividing a polynomial with another polynomial, the polynomial is written in standard class i.due east. the terms of the dividend and the divisor are arranged in decreasing order of their degrees. The method to solve these types of divisions is "Long sectionalisation". In algebra, an algorithm for dividing a polynomial past another polynomial of the same or lower degree is called polynomial long partitioning. It is the generalised version of the familiar arithmetic technique called long division. Permit united states of america have an example.

Example: Divide x ² + 2x + 3x ³ + 5 by ane + 2x + x² .

Solution:

Let us arrange the polynomial to be divided in the standard form.

3x^three + tenⁱⁱ + 2x + 5

Divisor = x² + 2x + i

Using the method of long partitioning of polynomials, let us divide 3x³ + x² + 2x + 5 by 10² + 2x + 1.

Step 1: To obtain the first term of the quotient, divide the highest caste term of the dividend, i.e. 3x³ by the highest degree term of the divisor, i.e. ten² .

3x³ /x² = 3x

Now, deport out the division process.

Step 2: Now, to obtain the 2d term of the quotient, dissever the highest degree term of the new dividend, i.eastward. –5x² by the highest degree term of the divisor, i.e. x² .

-5x² /10² = -5

Again carry out the segmentation process with – 5x^two – 10 + v (the residue in the previous step).

Footstep three: The remainder obtained from the previous step is 9x + x.

The caste of 9x + ten is less than the divisor x² + 2x + one. So, we cannot continue the division any further.

Dividing polynomial by another polynomial

Polynomial Division Algorithm

If p(x) and 1000(x) are any ii polynomials with g(10) ≠ 0, and then we can find polynomials q(x) and r(10) such that

p(ten) = g(x) × q(x) + r(ten)

Here,

r(x) = 0 or degree of r(x) < degree of g(x)

This result is called the Division Algorithm for polynomials.

From the previous example, we can verify the polynomial division algorithm as:

p(x) = 3x³ + x² + 2x + 5

g(10) = x^two + 2x + 1

Also, quotient = q(x) = 3x – i

remainder = r(ten) = 9x + x

Now,

g(x) × q(x) + r(x) = (x² + 2x + 1) × (3x – 5) + (9x + 10)

= 3x³ + 6x² + 3x – 5x² – 10x – 5 + 9x + 10

= 3x³ + xⁱⁱ + 2x + 5

= p(10)

Hence, the division algorithm is verified.

Polynomial Sectionalisation Questions

If the polynomial 10 ^iv – 6x³ + 16xⁱⁱ – 25x + 10 is divided past some other polynomial x² – 2x + k, the residue comes out to be x + a, find thousand and a.
Divide the polynomial 2t^four + 3t³ – 2t^two – 9t – 12 by t² – iii.
Find all the zeroes of 2x⁴ – 3xⁱⁱⁱ – 3x^two + 6x – 2, if two of its zeroes are √ii and −√2.