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Polynomial Fractions

Polynomial Fraction is an expression that is the ratio of two polynomials, like $\frac{P(x)}{Q(x)}$. To avoid number which make denominator zero (Q(x) $\neq$ 0), such as division by zero, which create unreal numbers. Polynomial fractions have same properties as those for numerical fractions, the only difference being that the numerator and denominator are both polynomials. We can multiply, divide, add and subtract polynomial fractions in the same way that we do in simple arithmetic.

A polynomial fraction consists of one polynomial divided by another polynomial.

Let P(x) and Q(x) be any two polynomials and Q(x) $\neq$ 0 then

F(x) = $\frac{P(x)}{Q(x)}$, Q(x) $\neq$ 0

Polynomial fraction is the ratio of two polynomials P(x) and Q(x), Except that Q(x) cannot be zero.

 Related Calculators Calculating Polynomials Polynomial Factor Mixed Fraction to Improper Fraction Add Polynomials Calculator

Simplifying Polynomial Fractions

Simplification of polynomial fractions with polynomials in the numerator or denominator can be simplified by factoring and reducing to the lowest terms.
To simplify a polynomial faction, start by factoring both the numerators and denominators completely.
Step for Simplifying Polynomial Fractions:

Step 1:
Factor the polynomial in the numerator or the denominator.

Step 2: Reduce the fraction to lowest terms by canceling out common
monomials or polynomials exist in both the numerator and denominator.

Step 3: Multiplying any monomials or polynomials that remain in the numerator or denominator.

Solved Examples

Question 1: Simplify the fraction $\frac{xy + 2y}{x^2y + 7yx + 10y}$
Solution:

Given $\frac{xy + 2y}{x^2y + 7yx + 10y}$

Step 1: Factorize the polynomials

xy + 2y = y(x + 2)

x2 y + 7xy + 10y = y(x2 + 7x + 10) = y(x + 2)(x + 5)

$\frac{xy + 2y}{x^2y + 7yx + 10y}$ = $\frac{y(x + 2)}{y(x + 2)(x + 5)}$

Step 2: Cancel common monomial exist in both the numerator and denominator

$\frac{xy + 2y}{x^2y + 7yx + 10y}$ = $\frac{y(x + 2)}{y(x^2 + 7x + 10)}$ = $\frac{y(x + 2)}{y(x + 2)(x + 5)}$ = $\frac{1}{x + 5}$.

Question 2: Simplify the fraction $\frac{x^2y - 49y}{x^2 + 6x - 7}$

Solution:

Given $\frac{x^2y - 49y}{x^2 + 6x - 7}$

Step 1: Factor the polynomials

x2 - 49 = x2 - 72 =  (x - 7)(x - 7)

x2 + 6x - 7 = (x + 7)(x - 1)

Step 2:  Cancel common monomial exist in both the numerator and denominator

$\frac{x^2y - 49y}{x^2 + 6x - 7}$ = $\frac{y(x^2 - 7^2)}{x^2 + 6x - 7}$ = $\frac{y(x - 7)(x + 7)}{(x + 7)(x - 1)}$

=
$\frac{y(x - 7)}{x - 1}$

Step 3:  Multiply remaining numerator and denominator

$\frac{y(x - 7)}{x - 7}$ = $\frac{xy - 7y}{x - 7}$

$\frac{x^2y - 49y}{x^2 + 6x - 7}$ = $\frac{xy - 7y}{x - 7}$

Multiplying Polynomial Fractions

Polynomial Fractions are multiply in the same way as the algebraic fractions. To multiply polynomials, factor both the numerator and denominator of both
fractions and multiply remaining polynomials.

Steps for Multiplying Polynomial Fractions:

Step 1: Factorize both the numerators and denominators of both the fractions.

Step 2: Cancel the factors common to the numerator and denominator of both the fractions.

Step 3: Multiply the remaining numerators together and denominators together.

Solved Examples

Question 1: Multiply $\frac{x^2y + 2y^2}{x^2 - 1}$ and $\frac{x + 1}{x^2 + 2y}$
Solution:

Given $\frac{x^2y + 2y^2}{x^2 - 1}$ $\times$ $\frac{x + 1}{x^2 + 2y}$

Step 1: Factorize the polynomials

x2y + 2y2 = y(x2 + 2y)

x2 - 1 = x2 - 12 = (x - 1)(x + 1)

Step 2: Cancel the common factors

$\frac{x^2y + 2y^2}{x^2 - 1}$ $\times$ $\frac{x + 1}{x^2 + 2y}$  = $\frac{y(x^2 + 2y)}{(x - 1)(x + 1)}$ $\times$ $\frac{x + 1}{x^2 + 2y}$ = $\frac{y}{x - 1}$.

Question 2: Multiply and simplify $\frac{xy + 2y}{x^2 - 100}$ and $\frac{x - 10}{xy - y}$
Solution:

Given $\frac{xy + 2y}{x^2 - 100}$ $\times$ $\frac{x - 10}{xy - y}$

Step 1: Factorize the polynomials

xy + 2y = y(x + 2)

x2 - 100 = (x2 - 102) = (x - 10)(x + 10)

xy - y = y(x - 1)

Step 2: Cancel the common factors

$\frac{xy + 2y}{x^2 - 100}$  $\times$ $\frac{x - 10}{xy - y}$ = $\frac{y(x + 2)}{(x - 10)(x + 10)}$  $\times$ $\frac{x - 10}{y(x - 1)}$

= $\frac{x + 2}{(x + 10)(x - 1)}$

Step 3:  Multiply remaining numerator and denominator of both the fractions

$\frac{x + 2}{(x + 10)(x - 1)}$ = $\frac{x + 2}{x^2 + 9x - 10}$

$\frac{xy + 2y}{x^2 - 100}$ $\times$ $\frac{x - 10}{xy - y}$  = $\frac{x + 2}{x^2 + 9x - 10}$

Dividing Polynomial Fractions

When dividing polynomial fractions, firstly invert the second fraction and then multiply. We can cancel only after we invert. Division also describe as to divide by a fraction, multiply by its reciprocal.

Steps for Dividing Algebraic Fractions:

Step 1: Invert the second fraction and change the division sign ($\div$) to a multiplication sign ($\times$).

Step 2: Factorize the numerators and denominators of both the fractions.

Step 3: Cancel the factors common to both the numerator and denominator.

Step 4: Multiply the remaining numerators together and denominators together.

Solved Examples

Question 1: Divide and simplify $\frac{x(x + 2)}{x^2 - 1}$ and $\frac{x}{x - 1}$
Solution:

Given $\frac{x(x + 2)}{x^2 - 1}$ $\div$ $\frac{x}{x - 1}$

Step 1: Factorize the polynomials

x2 - 1 = (x2 - 12) = (x - 1)(x + 1)

Step 2: Invert the second fraction and change divide sign($\div$) to multiply sign($\times$)

$\frac{x(x + 2)}{x^2 - 1}$ $\div$  $\frac{x}{x - 1}$ = $\frac{x(x + 2)}{x^2 - 1}$ $\times$  $\frac{x - 1}{x}$

Step 3: Cancel the common factors

$\frac{x(x + 2)}{(x - 1)(x + 1)}$ $\times$  $\frac{x - 1}{x}$ = $\frac{x + 2}{x + 1}$

Question 2: Divide and simplify $\frac{2x + 18}{x^2 + x}$ and $\frac{5x + 45}{x + 1}$
Solution:

Given $\frac{2x + 18}{x^2 + x}$  $\div$ $\frac{5x + 45}{x + 1}$

Step 1: Factorize the polynomials

2x + 18 = 2(x+ 9)

x2 + x = x(x + 1)

5x + 45 = 5(x + 9)

Step 2: Invert the second fraction and change divide sign($\div$) to multiply sign($\times$)

$\frac{2x + 18}{x^2 + x}$  $\div$ $\frac{5x + 45}{x + 1}$

= $\frac{2x + 18}{x^2 + x}$  $\times$ $\frac{x + 1}{5x + 45}$

Step 3: Cancel the common factors

$\frac{2x + 18}{x^2 + x}$  $\times$ $\frac{x + 1}{5x + 45}$ = $\frac{2(x + 9)}{x(x + 1)}$  $\times$ $\frac{x + 1}{5(x + 9)}$

= $\frac{2}{5x}$

⇒  $\frac{2x + 18}{x^2 + x}$  $\div$ $\frac{5x + 45}{x + 1}$  = $\frac{2}{5x}$

Question 3: Solve $\frac{xy}{xy^2 + xy}$ $\div$ $\frac{x^2}{y + 1}$
Solution:

Given $\frac{xy}{xy^2 + xy}$ $\div$ $\frac{x^2}{y + 1}$

Step 1: Factorize the polynomials

xy2 + xy = xy(y + 1)

Step 2: Invert the second fraction and change divide sign($\div$) to multiply sign($\times$)

$\frac{xy}{xy^2 + xy}$ $\div$ $\frac{x^2}{y + 1}$ = $\frac{xy}{xy^2 + xy}$ $\times$ $\frac{y + 1}{x^2}$

Step 3: Cancel the common factors

$\frac{xy}{xy^2 + xy}$ $\times$ $\frac{y + 1}{x^2}$ = $\frac{xy}{xy(y + 1)}$ $\times$ $\frac{y + 1}{x^2}$ = $\frac{1}{x^2}$

⇒   $\frac{xy}{xy^2 + xy}$ $\div$ $\frac{x^2}{y + 1}$   = $\frac{1}{x^2}$

Adding of polynomial fractions, like number fractions, having common denominators. When the fraction to be added do not have common denominators, they must be converted to create.

Step 1:
Find a lowest common denominator.

Step 2:
Change each fraction to an equivalent fraction with the common denominator.

Step 3:
Combine numerators and reduce if possible.

Solved Examples

Question 1: Solve $\frac{x^2 + 1}{2x + 1}$ + $\frac{x^2 +3x}{2x + 1}$
Solution:

Given $\frac{x^2 + 1}{2x + 1}$ + $\frac{x^2 +3x}{2x + 1}$

Step 1:

$\frac{x^2 + 1}{2x + 1}$ + $\frac{x^2 +3x}{2x + 1}$ = $\frac{x^2 + 1 + x^2 + 3x}{2x + 1}$

=
$\frac{2x^2 + 3x + 1}{2x + 1}$

Step 2:

$\frac{x^2 + 1}{2x + 1}$ + $\frac{x^2 +3x}{2x + 1}$ = $\frac{2x^2 + 3x + 1}{2x + 1}$

=
$\frac{(x + 1)(2x + 1)}{2x + 1}$ = x + 1

$\frac{x^2 + 1}{2x + 1}$ + $\frac{x^2 +3x}{2x + 1}$ = x + 1

Question 2: Add $\frac{xy}{x + 2}$ and $\frac{x^2}{y}$
Solution:

Given  $\frac{xy}{x + 2}$ $\frac{x^2}{y}$

LCM of x + 2 and y is y(x + 2)

$\frac{xy}{x + 2}$ $\frac{x^2}{y}$ = $\frac{xy\times y + x^2(x + 2)}{y(x + 2)}$

=
$\frac{xy^2 + x^3 + 2x^2}{y(x + 2)}$

$\frac{xy^2 + x^3 + 2x^2}{yx + 2y}$.

Question 3: Solve $\frac{2x + 3}{x + 1}$ $\frac{x - 1}{3x}$
Solution:

Given $\frac{2x + 3}{x + 1}$ $\frac{x - 1}{3x}$

LCM of  x + 1 and 3x is 3x(x + 1)

$\frac{2x + 3}{x + 1}$ $\frac{x - 1}{3x}$ = $\frac{(2x + 3)3x + (x - 1)(x + 1)}{3x(x + 1)}$

= $\frac{6x^2 + 9x + x^2 - 1}{3x(x + 1)}$

=
$\frac{7x^2 + 9x - 1}{3x(x + 1)}$

$\frac{2x + 3}{x + 1}$ $\frac{x - 1}{3x}$ =  $\frac{7x^2 + 9x - 1}{3x^2 + 3x}$.

Subtracting Polynomial Fractions

To subtract Polynomial fractions in the same way that we do in simple arithmetic. The following steps can be used for the subtraction of polynomial fractions.

Steps for Subtracting Polynomial Fractions:

Step 1:
Find a lowest common denominator.

Step 2:
Change each fraction to an equivalent fraction with the common denominator.

Step 3:
Subtract numerators and reduce if possible.

Solved Examples

Question 1: Subtract $\frac{x + 1}{x}$ from $\frac{x^2 + 2}{x^2 - 3}$
Solution:

Given $\frac{x^2 + 2}{x^2 - 3}$ - $\frac{x + 1}{x}$

LCM of x2 - 3 and x is x(x2 - 3)

$\frac{x^2 + 2}{x^2 - 3}$ - $\frac{x + 1}{x}$ = $\frac{x(x^2 + 2) - (x + 1)(x^2 - 3)}{x(x^2 - 3)}$

=
$\frac{x^3 + 2x - x^3 - x^2 + 3x + 3}{x(x^2 - 3)}$

=
$\frac{- x^2 + 5x + 3}{x(x^2 - 3)}$

$\frac{x^2 + 2}{x^2 - 3}$ - $\frac{x + 1}{x}$ = $\frac{- x^2 + 5x + 3}{x(x^2 - 3)}$.

Question 2: Solve $\frac{x + 2}{x}$ - $\frac{x - 2}{x + 1}$
Solution:

Given $\frac{x + 2}{x}$ - $\frac{x - 2}{x + 1}$

LCM of x and x +1 is x(x + 1)

$\frac{x + 2}{x}$ - $\frac{x - 2}{x + 1}$ = $\frac{(x + 2)(x + 1) - x(x - 2)}{x(x + 1)}$

= $\frac{x^2 + 2x + x + 2 - x^2 + 2x}{x(x + 1)}$

= $\frac{5x + 2}{x(x + 1)}$

Question 3: Solve $\frac{2m}{m + 1}$ - $\frac{5n}{n + 1}$

Solution:

Given $\frac{2m}{m + 1}$ - $\frac{5n}{n + 1}$

LCM of m + 1 and n + 1 is (m +1)(n + 1)

$\frac{2m}{m + 1}$ - $\frac{5n}{n + 1}$ = $\frac{2m(n + 1) - 5n(m + 1)}{(m + 1)(n +1)}$

=
$\frac{2mn + 2m - 5mn - 5n}{(m + 1)(n +1)}$

=
$\frac{- 3mn + 2m - 5n}{(m + 1)(n +1)}$

$\frac{2m}{m + 1}$ - $\frac{5n}{n + 1}$ = $\frac{- 3mn + 2m - 5n}{(m + 1)(n +1)}$

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