Question 1: Identify monomials, binomials and trinomials from the following:

\displaystyle \text{i) } \frac{1}{2} a^2b^2c^2            \displaystyle \text{ii) } 7x \times y^2 \times z^2            \displaystyle \text{iii) } \frac{9x^3}{z}            \displaystyle \text{iv) }2x+5

\displaystyle \text{v) }\frac{a}{3} + \frac{b}{6}            \displaystyle \text{vi) }\text{i) } xy+yz+zx             \displaystyle \text{vii) } \frac{x^2-2y^2+3z^2}{3}

\displaystyle \text{viii) } 8a\div{}9b-2a^2 \times b^2            \displaystyle \text{ix) } 7x^2+ \frac{2y^2+1}{5}  

Answer:

\displaystyle \text{i) } \frac{1}{2} a^2b^2c^2 \text{ is a Monomial }

\displaystyle \text{ii) } 7x \times y^2 \times z^2 \text{ is a Monomial }

\displaystyle \text{iii) } \frac{9x^3}{z} \text{ is a Monomial }

\displaystyle \text{iv) } 2x+5 \text{ is a Binomial }

\displaystyle \text{v) } \frac{a}{3} + \frac{b}{6} \text{ is a Binomial }

\displaystyle \text{vi) } xy+yz+zx \text{ is a Trinomial }

\displaystyle \text{vii) } \frac{x^2-2y^2+3z^2}{3} \text{ is a Trinomial }

\displaystyle \text{viii) } 8a\div{}9b-2a^2 \times b^2 \text{ is a Binomial }

\displaystyle \text{ix) } 7x^2+ \frac{2y^2+1}{5} \text{ is a Trinomial }

 \displaystyle \\

Question 2: Write the numerical and literal coefficient of each of the following:

\displaystyle \text{i) } -7x^2y           \displaystyle \text{ii) } \pi{}r^2 \displaystyle \text{iii) } \frac{2a}{3}           \displaystyle \text{iv) }  5a^2 \times b \div 2c           \displaystyle \text{v) }  \frac{-7pq}{9xy}  

Answer:

\displaystyle \text{i) } -7x^2y \text{ Numerical coefficient } : -7 \text{ Literal coefficient } : x^2 y

\displaystyle \text{ii) } \pi{}r^2 \text{ Numerical coefficient } : \pi \text{ Literal coefficient } : x^2 y

\displaystyle \text{iii) } \frac{2a}{3} \text{ Numerical coefficient } : \frac{2}{3} \text{ Literal coefficient } : a

\displaystyle \text{iv) }  5a^2 \times b \div 2c \text{ Numerical coefficient } : \frac{5}{2} \text{ Literal coefficient } : a^2b\div{}2c

\displaystyle \text{v) }  \frac{-7pq}{9xy} \text{ Numerical coefficient } : \frac{-7}{9} \text{ Literal coefficient } : \frac{pq}{xy}  

 \displaystyle \\

Question 3: In, \displaystyle - \frac{3}{5} x^3y^2z write down the coefficient for:

\displaystyle \text{i) } x^2            \displaystyle \text{ii) } -yz            \displaystyle \text{iii) } \frac{3}{5} xyz             \displaystyle \text{ iv) }  {-x}^2y

Answer:

\displaystyle \text{i) } \text{Coefficient for } x^2 : - \frac{3}{5} xy^2z            \displaystyle \text{ii) } \text{Coefficient for } -yz : + \frac{3}{5} x^3y

\displaystyle \text{iii) } \text{Coefficient for } + \frac{3}{5} xyz : -x^2y            \displaystyle \text{ iv) }  \text{Coefficient for } -x^2y : \frac{3}{5} xyz

 \displaystyle \\

Question 4: Identify the pairs of like terms:

\displaystyle \text{i) } \frac{x}{2} ,- \frac{x}{3}           \displaystyle \text{ii) } {6a}^2bc,6ab^2c           \displaystyle \text{iii) } 6pq,-3qx          \displaystyle \text{ iv) }  8a^2,- \frac{2}{3} a^2           \displaystyle \text{ v) }  2x,2y           \displaystyle \text{ vi) }  {3xy}^2p,-8py^2x

Answer:

\displaystyle \text{i) } \frac{x}{2} ,- \frac{x}{3} : \text{ The two terms are like terms. }

\displaystyle \text{ii) } {6a}^2bc,6ab^2c : \text{ The two terms are unlike terms. }

\displaystyle \text{iii) } 6pq,-3qx : \text{ The two terms are unlike terms. }

\displaystyle \text{ iv) }  8a^2,- \frac{2}{3} a^2 : \text{ The two terms are like terms. }

\displaystyle \text{ v) }  2x,2y : \text{ The two terms are unlike terms. }

\displaystyle \text{ vi) } {3xy}^2p,-8py^2x : \text{ The two terms are like terms. }

 \displaystyle \\

Question 5: Which of the following expressions are polynomials?

\displaystyle \text{i) } 1-x            \displaystyle \text{ii) } 3+y+y^2          \displaystyle \text{iii) } z+\sqrt{z}          \displaystyle \text{ iv) }  x- \frac{1}{x}  

\displaystyle \text{ v) }  x^3+x\sqrt{x}-x+2          \displaystyle \text{ vi) }  x^2+y^2+xy+x^2y^2          \displaystyle \text{ vii) }  5  

\displaystyle \text{viii) } \frac{1}{3} x^3-x^4          \displaystyle \text{ix) } x^2+\sqrt{3}x+5          \displaystyle \text{x) } {5x}^2+6xy-7\sqrt{y}  

\displaystyle \text{xi) } 6x^2\sqrt{y}-3xy+5

Answer:

\displaystyle \text{i) } 1-x : \text{ Is a polynomial. }

\displaystyle \text{ii) } 3+y+y^2 : \text{ Is a polynomial. }

\displaystyle \text{iii) } z+\sqrt{z} : \text{ Is Not a polynomial. }

\displaystyle \text{ iv) } x- \frac{1}{x} : \text{ Is Not a polynomial. }

\displaystyle \text{ v) } x^3+x\sqrt{x}-x+2 : \text{ Is Not a polynomial. }

\displaystyle \text{ vi) } x^2+y^2+xy+x^2y^2 : \text{ Is a polynomial. }

\displaystyle \text{vii) } 5 : \text{ Is a polynomial. }

\displaystyle \text{viii) } \frac{1}{3} x^3-x^4 : \text{ Is a polynomial. }

\displaystyle \text{ix) } x^2+\sqrt{3}x+5 : \text{ Is a polynomial. }

\displaystyle \text{x) } {5x}^2+6xy-7\sqrt{y} : \text{ Is Not a polynomial. }

\displaystyle \text{xi) } 6x^2\sqrt{y}-3xy+5 : \text{ Is Not a polynomial. }

 \displaystyle \\

Question 6: Write the degree of each of the following polynomials:

\displaystyle \text{i) } 2-x          \displaystyle \text{ii) } 3-x^2+x^3          \displaystyle \text{iii) } {5x}^2-6x          \displaystyle \text{ iv) } {2x}^3-8x          

\displaystyle \text{ v) } 1-x+x^4-{3x}^2           \displaystyle \text{ vi) }  z^3-z^4+{2z}^2-6          \displaystyle \text{ vii) } 1-y-y^2+{3y}^5          

\displaystyle \text{viii) } x^2- \frac{x}{2}            \displaystyle \text{ix) } t^4-t^3+2t-{3t}^6          \displaystyle \text{x) } 5          \displaystyle \text{xi) } 9-x^2          \displaystyle \text{xii) } 1-x^3

Answer:

\displaystyle \text{i) } 2-x : \text{ degree }   1

\displaystyle \text{ii) } 3-x^2+x^3 : \text{ degree }    3  

\displaystyle \text{iii) } {5x}^2-6x : \text{ degree }    2

i\displaystyle \text{ iv) } {2x}^3-8x : \text{ degree }    3  

i\displaystyle \text{ v) }1-x+x^4-{3x}^2 : \text{ degree }    4

i\displaystyle \text{ vi) }  z^3-z^4+{2z}^2-6 : \text{ degree }    4

i\displaystyle \text{ vii) }  1-y-y^2+{3y}^5 : \text{ degree }    5

\displaystyle \text{viii) } x^2- \frac{x}{2} : \text{ degree }    2  

\displaystyle \text{ix) } t^4-t^3+2t-{3t}^6 : \text{ degree }    6

\displaystyle \text{x) } 5 : \text{ degree }    0

\displaystyle \text{xi) } 9-x^2 : \text{ degree }    2

\displaystyle \text{xii) } 1-x^3 : \text{ degree }    3

 \displaystyle \\

Question .7. Write the degree of each of the following polynomials:

\displaystyle \text{i) } xy+yz+zx+3xyz            \displaystyle \text{ii) } a^2+b^2+c^2-3abc

\displaystyle \text{iii) } 2xy+3xy^2+{5x}^2y+{7x}^2y^2              \displaystyle \text{iv) } a^5-b^5-{2a}^3b^3

\displaystyle \text{v) } x^2y+{xy}^2+5xy            \displaystyle \text{vi) } 1+2x+5x^2y+{6yz}^2

Answer:

\displaystyle \text{i) } xy+yz+zx+3xyz :  \text{ The degree of the polynomial is }   3

\displaystyle \text{ii) } a^2+b^2+c^2-3abc : \text{ The degree of the polynomial is } 3

\displaystyle \text{iii) } 2xy+3xy^2+{5x}^2y+{7x}^2y^2 : \text{ The degree of the polynomial is } 4

iv) \displaystyle a^5-b^5-{2a}^3b^3 : \text{ The degree of the polynomial is } 6

v) \displaystyle x^2y+{xy}^2+5xy : \text{ The degree of the polynomial is } 3

v\displaystyle \text{i) } 1+2x+5x^2y+{6yz}^2 : \text{ The degree of the polynomial is } 3

 \displaystyle \\

Question 8: Explain the following:

\displaystyle \text{i) } \text{Find the value of } {4x}^3-{3x}^2+5x-6 \text{ when } x=4

\displaystyle \text{ii) } \text{Find the value of } ^3-8x^2+14x-7 \text{ when } x=3

Answer:

\displaystyle \text{i) } \text{Find the value of } {4x}^3-{3x}^2+5x-6 \text{ when } x=4

 \displaystyle {4x}^3-3x^2+5x-6 = 4 \times 4^3-3 \times 4^2+5 \times 4-6 =90

\displaystyle \text{ii) } \text{Find the value of } ^3-8x^2+14x-7 \text{ when } x=3

 \displaystyle x^3-8x^2+14x-7= 3^3-8{ \times 3}^2+14 \times 3-7 =-30

 \displaystyle \\

Question 9: If \displaystyle a=4 and \displaystyle b=5 , \text{Find the value of } a^3+b^3-{3a}^2b+{3ab}^2

Answer:

 \displaystyle a^3+b^3-{3a}^2b+{3ab}^2 = 4^3+5^3-{3 \times 4}^2 \times 5+{3 \times 4 \times 5}^2 = 249

 \displaystyle \\

Question 10: If \displaystyle x=4, y=3 and \displaystyle z=-2 , find the value of

\displaystyle \text{i) } x^2+y^2+z^2+2xyz \displaystyle \text{ii) } x^3+y^3+z^3-3xyz

Answer:

\displaystyle \text{i) } x^2+y^2+z^2+2xyz = 4^2+3^2+({-2)}^2+2xyz = -19

$latex \displaystyle \text{ii) } x^3+y^3+z^3-3xyz = 4^3+3^3+{(-2)}^3-3\left(4\right)\left(3\right)(-2) = 155