\displaystyle \textbf{Question 1: }\text{Identify the monomials, binomials and trinomials}
\displaystyle \text{from the following expressions:}
\displaystyle \text{(i) }\frac{1}{2}a^2b^2c^2\qquad \text{(ii) }7xy^2z^2\qquad\text{(iii) }\frac{9x^3}{z}
\displaystyle \text{(iv) }2x+5\qquad \text{(v) }\frac{a}{3}+\frac{b}{6}\qquad\text{(vi) }xy+yz+zx
\displaystyle \text{(vii) }\frac{x^2-2y^2+3z^2}{3}\qquad \text{(viii) }\frac{8a}{9b}-2a^2b^2
\displaystyle \text{(ix) }7x^2+\frac{2y^2+1}{5}
\displaystyle \text{Answer:}
\displaystyle \text{(i) }\frac{1}{2}a^2b^2c^2\text{ is a monomial}.

\displaystyle \text{(ii) }7xy^2z^2\text{ is a monomial}.

\displaystyle \text{(iii) }\frac{9x^3}{z}\text{ is a monomial algebraic expression}.

\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} =\frac{x^2}{3}-\frac{2y^2}{3}+z^2\text{ is a trinomial}.

\displaystyle \text{(viii) }\frac{8a}{9b}-2a^2b^2\text{ is a binomial}.

\displaystyle \text{(ix) }7x^2+\frac{2y^2+1}{5} =7x^2+\frac{2y^2}{5}+\frac{1}{5}\text{ is a trinomial}.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Write the numerical and literal coefficients of each}
\displaystyle \text{of the following:}
\displaystyle \text{(i) }-7x^2y\qquad \text{(ii) }\pi r^2\qquad\text{(iii) }\frac{2a}{3}
\displaystyle \text{(iv) }\frac{5a^2b}{2c}\qquad \text{(v) }-\frac{7pq}{9xy}
\displaystyle \text{Answer:}
\displaystyle \text{(i) }-7x^2y:\quad \text{Numerical coefficient}=-7,\quad\text{Literal coefficient}=x^2y.

\displaystyle \text{(ii) }\pi r^2:\quad \text{Numerical coefficient}=\pi,\quad\text{Literal coefficient}=r^2.

\displaystyle \text{(iii) }\frac{2a}{3}:\quad \text{Numerical coefficient}=\frac{2}{3},\quad\text{Literal coefficient}=a.

\displaystyle \text{(iv) }\frac{5a^2b}{2c}:\quad \text{Numerical coefficient}=\frac{5}{2},\quad \text{Literal coefficient}=\frac{a^2b}{c}.

\displaystyle \text{(v) }-\frac{7pq}{9xy}:\quad \text{Numerical coefficient}=-\frac{7}{9},\quad \text{Literal coefficient}=\frac{pq}{xy}.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{In }-\frac{3}{5}x^3y^2z,\text{ write down the coefficient of:}
\displaystyle \text{(i) }x^2\qquad\text{(ii) }-yz\qquad \text{(iii) }\frac{3}{5}xyz\qquad\text{(iv) }-x^2y
\displaystyle \text{Answer:}
\displaystyle \text{(i) Coefficient of }x^2=-\frac{3}{5}xy^2z.

\displaystyle \text{(ii) Coefficient of }-yz=\frac{3}{5}x^3y.

\displaystyle \text{(iii) Coefficient of }\frac{3}{5}xyz=-x^2y.

\displaystyle \text{(iv) Coefficient of }-x^2y=\frac{3}{5}xyz.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Identify whether the following pairs are like or unlike terms:}
\displaystyle \text{(i) }\frac{x}{2},-\frac{x}{3}\qquad \text{(ii) }(6a)^2bc,\ 6ab^2c\qquad\text{(iii) }6pq,\ -3qx
\displaystyle \text{(iv) }8a^2,-\frac{2}{3}a^2\qquad \text{(v) }2x,\ 2y\qquad\text{(vi) }(3xy)^2p,\ -8py^2x
\displaystyle \text{Answer:}
\displaystyle \text{(i) }\frac{x}{2},-\frac{x}{3}:\quad\text{The two terms are like terms}.

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

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

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

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

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

\displaystyle \textbf{Question 5: }\text{Which of the following expressions are polynomials?}
\displaystyle \text{(i) }1-x\qquad\text{(ii) }3+y+y^2\qquad \text{(iii) }z+\sqrt{z}\qquad\text{(iv) }x-\frac{1}{x}
\displaystyle \text{(v) }x^3+x\sqrt{x}-x+2\qquad \text{(vi) }x^2+y^2+xy+x^2y^2\qquad\text{(vii) }5
\displaystyle \text{(viii) }\frac{1}{3}x^3-x^4\qquad \text{(ix) }x^2+\sqrt{3}x+5
\displaystyle \text{(x) }(5x)^2+6xy-7\sqrt{y}\qquad \text{(xi) }6x^2\sqrt{y}-3xy+5
\displaystyle \text{Answer:}
\displaystyle \text{(i) }1-x:\quad\text{It is a polynomial}.

\displaystyle \text{(ii) }3+y+y^2:\quad\text{It is a polynomial}.

\displaystyle \text{(iii) }z+\sqrt{z}:\quad\text{It is not a polynomial since }\sqrt{z}=z^{1/2}.

\displaystyle \text{(iv) }x-\frac{1}{x}:\quad\text{It is not a polynomial since }\frac{1}{x}=x^{-1}.

\displaystyle \text{(v) }x^3+x\sqrt{x}-x+2:\quad \text{It is not a polynomial since }x\sqrt{x}=x^{3/2}.

\displaystyle \text{(vi) }x^2+y^2+xy+x^2y^2:\quad\text{It is a polynomial}.

\displaystyle \text{(vii) }5:\quad\text{It is a constant polynomial}.

\displaystyle \text{(viii) }\frac{1}{3}x^3-x^4:\quad\text{It is a polynomial}.

\displaystyle \text{(ix) }x^2+\sqrt{3}x+5:\quad\text{It is a polynomial}.

\displaystyle \text{(x) }(5x)^2+6xy-7\sqrt{y}:\quad \text{It is not a polynomial since }\sqrt{y}=y^{1/2}.

\displaystyle \text{(xi) }6x^2\sqrt{y}-3xy+5:\quad \text{It is not a polynomial since }\sqrt{y}=y^{1/2}.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Write the degree of each of the following polynomials:}
\displaystyle \text{(i) }2-x\qquad\text{(ii) }3-x^2+x^3\qquad \text{(iii) }(5x)^2-6x
\displaystyle \text{(iv) }(2x)^3-8x\qquad \text{(v) }1-x+x^4-(3x)^2
\displaystyle \text{(vi) }z^3-z^4+(2z)^2-6\qquad \text{(vii) }1-y-y^2+(3y)^5
\displaystyle \text{(viii) }x^2-\frac{x}{2}\qquad \text{(ix) }t^4-t^3+2t-(3t)^6
\displaystyle \text{(x) }5\qquad \text{(xi) }9-x^2\qquad\text{(xii) }1-x^3
\displaystyle \text{Answer:}
\displaystyle \text{(i) }2-x:\quad\text{Degree}=1.

\displaystyle \text{(ii) }3-x^2+x^3:\quad\text{Degree}=3.

\displaystyle \text{(iii) }(5x)^2-6x:\quad\text{Degree}=2.

\displaystyle \text{(iv) }(2x)^3-8x:\quad\text{Degree}=3.

\displaystyle \text{(v) }1-x+x^4-(3x)^2:\quad\text{Degree}=4.

\displaystyle \text{(vi) }z^3-z^4+(2z)^2-6:\quad\text{Degree}=4.

\displaystyle \text{(vii) }1-y-y^2+(3y)^5:\quad\text{Degree}=5.

\displaystyle \text{(viii) }x^2-\frac{x}{2}:\quad\text{Degree}=2.

\displaystyle \text{(ix) }t^4-t^3+2t-(3t)^6:\quad\text{Degree}=6.

\displaystyle \text{(x) }5:\quad\text{Degree}=0.

\displaystyle \text{(xi) }9-x^2:\quad\text{Degree}=2.

\displaystyle \text{(xii) }1-x^3:\quad\text{Degree}=3.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Write the degree of each of the following polynomials:}
\displaystyle \text{(i) }xy+yz+zx+3xyz\qquad \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\qquad \text{(v) }x^2y+(xy)^2+5xy
\displaystyle \text{(vi) }1+2x+5x^2y+(6yz)^2
\displaystyle \text{Answer:}
\displaystyle \text{(i) }xy+yz+zx+3xyz:\quad \text{The degree of the polynomial is }3.

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

\displaystyle \text{(iii) }2xy+3xy^2+(5x)^2y+(7x)^2y^2:\quad \text{The degree is }4.

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

\displaystyle \text{(v) }x^2y+(xy)^2+5xy:\quad \text{The degree of the polynomial is }4.

\displaystyle \text{(vi) }1+2x+5x^2y+(6yz)^2:\quad \text{The degree of the polynomial is }4.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Evaluate the following:}
\displaystyle \text{(i) }4x^3-3x^2+5x-6\text{ when }x=4
\displaystyle \text{(ii) }x^3-8x^2+14x-7\text{ when }x=3
\displaystyle \text{Answer:}
\displaystyle \text{(i) Substituting }x=4,\text{ we get}
\displaystyle 4x^3-3x^2+5x-6 =4(4)^3-3(4)^2+5(4)-6.
\displaystyle =4(64)-3(16)+20-6=256-48+20-6=222.
\displaystyle \therefore \text{the value of the expression is }222.

\displaystyle \text{(ii) Substituting }x=3,\text{ we get}
\displaystyle x^3-8x^2+14x-7 =(3)^3-8(3)^2+14(3)-7.
\displaystyle =27-72+42-7=-10.
\displaystyle \therefore \text{the value of the expression is }-10.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{If }a=4\text{ and }b=5,\text{ find the value of}
\displaystyle a^3+b^3-3a^2b+3ab^2.
\displaystyle \text{Answer:}
\displaystyle \text{Substituting }a=4\text{ and }b=5,\text{ we get}
\displaystyle a^3+b^3-3a^2b+3ab^2
\displaystyle =(4)^3+(5)^3-3(4)^2(5)+3(4)(5)^2.
\displaystyle =64+125-240+300=249.
\displaystyle \therefore \text{the value of the expression is }249.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{If }x=4,\ y=3\text{ and }z=-2,\text{ find the value of:}
\displaystyle \text{(i) }x^2+y^2+z^2+2xyz\qquad \text{(ii) }x^3+y^3+z^3-3xyz
\displaystyle \text{Answer:}
\displaystyle \text{(i) Substituting }x=4,\ y=3\text{ and }z=-2,\text{ we get}
\displaystyle x^2+y^2+z^2+2xyz
\displaystyle =(4)^2+(3)^2+(-2)^2+2(4)(3)(-2).
\displaystyle =16+9+4-48=-19.
\displaystyle \therefore \text{the value of the expression is }-19.

\displaystyle \text{(ii) Substituting }x=4,\ y=3\text{ and }z=-2,\text{ we get}
\displaystyle x^3+y^3+z^3-3xyz
\displaystyle =(4)^3+(3)^3+(-2)^3-3(4)(3)(-2).
\displaystyle =64+27-8+72=155.
\displaystyle \therefore \text{the value of the expression is }155.
\displaystyle \\

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