Degree of the Polynomials
Category : 8th Class
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:
\[={{x}^{2}}+\frac{1}{{{x}^{2}}}+2-2x-\frac{2}{x}=\]\[{{\left( x+\frac{1}{x} \right)}^{2}}-2\left( x+\frac{1}{x} \right)=\] \[\left( x+\frac{1}{x} \right)\left( x+\frac{1}{x}-2 \right)\]
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)
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