8th Class Mathematics Factorisation Degree of the Polynomials

         Degree of the Polynomials

The degree of the polynomials is the highest of power of the variable in the given polynomials. If the degree of the polynomial is zero then it is called constant polynomial.

If the degree of the polynomial is one then it is called linear polynomial and if the degree of the polynomial is two then it is called quadratic polynomial. For cubic polynomial the degree is three and if the degree is four then it is called biquadrate polynomial.  

The polynomial \[ax+b=0\] is a linear polynomial.

The polynomial \[a{{x}^{2}}+bx+c=0\] is a quadratic polynomial.

The polynomial \[d{{x}^{3}}+a{{x}^{2}}+bx+c=0\] is a cubic polynomial.

The polynomial \[e{{x}^{4}}+d{{x}^{3}}+a{{x}^{2}}+bx+c=0\] is a biquadrate polynomial.  

Factories: \[{{\text{y}}^{\text{2}}}+\text{3y}+\text{y}+\text{3}\]

Solution:

\[={{y}^{2}}+3y+y+3=y(y+3)+1(y+3)=(y+3)(y+1)\]  

Factories: \[{{x}^{2}}+\frac{1}{{{x}^{2}}}+2-2x-\frac{2}{x}\]  

Solution:

\[={{x}^{2}}+\frac{1}{{{x}^{2}}}+2-2x-\frac{2}{x}=\]\[{{\left( x+\frac{1}{x} \right)}^{2}}-2\left( x+\frac{1}{x} \right)=\] \[\left( x+\frac{1}{x} \right)\left( x+\frac{1}{x}-2 \right)\]    

• A prime number has no factors other than 1 and itself.
• There are infinite number of prime numbers.
• The only prime number that is even is 2.
• 1 is not a prime number.
• Every composite number can be expressed as product of primes.    

• When we factories an expression, we write it as a product of factors.
• We can factories the expression by splitting the middle term if it is a quadratic equation.
• Division can be carried out through reducing the given expression or by merely performing the division.
• For division dividend \[=Divisor\times quotient+remainder.\] 
• Division can also be carried out by taking out the common factors and cancelling out the common terms in both numerators and denominators.  

  Factories \[4{{x}^{2}}+20x+3xy+15y\] and choose the correct option.

(a) \[(x+5)(4x+3y)\]                       

(b) \[(x-5)(5x+3y)\]

(c) \[(x+5)(5x-3y)\]                        

(d) \[(x+5)(x+3y)\]

(e) None of these

Answer: (a)

Explanation:

\[=(4{{x}^{2}}+20x)+(3yx+15y)\] \[=4x(x+5)+2y(x+5)\] \[=(x+5)(4x+3y)\]  

  Factories \[{{x}^{3}}-27\] and choose the correct option.

(a) \[(x-3)({{x}^{2}}+3x+9)\]                      

(b)\[(x+3)({{x}^{2}}+3x+9)\]

(c) \[(x-3)({{x}^{2}}-3x+9)\]                        

(d)\[(x-3)({{x}^{2}}+3x-9)\]

(e) None of these  

Answer: (a)

Explanation:

\[{{x}^{3}}-27=(x-3)({{x}^{2}}+3x+9)\]    

  Factories the given polynomial \[\frac{9}{16}{{x}^{2}}+\frac{4}{9}{{y}^{2}}+4{{z}^{2}}-xy-\frac{8}{3}yz+3zx\].

(a) \[{{\left( \frac{3}{4}x-\frac{2}{3}y+2z \right)}^{2}}\]                 

(b) \[{{\left( \frac{3}{4}x+\frac{2}{3}y+2z \right)}^{2}}\]

(c) \[{{\left( \frac{3}{4}x+\frac{2}{3}y-2z \right)}^{2}}\]                 

(d) \[{{\left( \frac{3}{4}x-\frac{2}{3}y-2z \right)}^{2}}\]

(e) None of these

Answer: (a)  

  Factories the given polynomial \[36{{u}^{2}}+\frac{1}{25}{{v}^{2}}+25{{w}^{2}}-\frac{12}{5}uv-\frac{8}{5}vw+48wu\].

(a) \[{{\left( 6u-\frac{v}{5}-5w \right)}^{2}}\]                     

(b) \[{{\left( 6u+\frac{v}{5}-5w \right)}^{2}}\]

(c) \[{{\left( 6u-\frac{v}{5}+5w \right)}^{2}}\]                    

(d) \[{{\left( 6u+\frac{v}{5}+5w \right)}^{2}}\]

(e) None of these

Answer: (c)  

  Simplify the given expression: \[=\frac{0.86\times 0.86\times 0.86+0.14\times 0.14\times 0.14}{0.86\times 0.86-0.86\times 0.14\times 0.14\times 0.14}\]

(a) 1                                                      

(b) 0

(c) 2                                                      

(d) -1

(e) None of these

Answer: (a)  

  Factories the expression given by \[18{{x}^{3}}{{y}^{3}}-27{{x}^{2}}{{y}^{3}}+36{{x}^{3}}{{y}^{2}}\]

(a) \[9{{x}^{2}}{{y}^{2}}(2xy-3y+4x)\]    

(b) \[9{{x}^{2}}{{y}^{2}}(2xy+3y+4x)\]

(c) \[9{{x}^{2}}{{y}^{2}}(2xy-3y+4x)\]                    

(d) \[9{{x}^{2}}{{y}^{2}}(2xy-3y-4x)\]

(e) None of these

Answer: (a)