\displaystyle \textbf{Question 1: }\text{Add:}
\displaystyle \text{(i) }2ab,\ -7ab,\ \frac{2}{3}ab,\ 9ab
\displaystyle \text{(ii) }5x^2y,\ -3x^2y,\ \frac{1}{2}x^2y,\ \frac{4}{5}x^2y
\displaystyle \text{Answer:}
\displaystyle \text{(i) }2ab+(-7ab)+\frac{2}{3}ab+9ab
\displaystyle =\left(2-7+\frac{2}{3}+9\right)ab =\frac{14}{3}ab.

\displaystyle \text{(ii) }5x^2y+(-3x^2y)+\frac{1}{2}x^2y+ \frac{4}{5}x^2y
\displaystyle =\left(5-3+\frac{1}{2}+\frac{4}{5}\right)x^2y =\frac{33}{10}x^2y.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Subtract:}
\displaystyle \text{(i) }7xy\text{ from }2xy\qquad \text{(ii) }2ab\text{ from }-3ab
\displaystyle \text{(iii) }-3a^2b\text{ from }2a^2b\qquad \text{(iv) }-5xy^2\text{ from }-3xy^2
\displaystyle \text{(v) }-\frac{3}{7}x^2y\text{ from } \frac{1}{3}x^2y
\displaystyle \text{Answer:}
\displaystyle \text{(i) }2xy-7xy=(2-7)xy=-5xy.

\displaystyle \text{(ii) }-3ab-2ab=(-3-2)ab=-5ab.

\displaystyle \text{(iii) }2a^2b-(-3a^2b) =(2+3)a^2b=5a^2b.

\displaystyle \text{(iv) }-3xy^2-(-5xy^2) =(-3+5)xy^2=2xy^2.

\displaystyle \text{(v) }\frac{1}{3}x^2y-\left(-\frac{3}{7}x^2y\right)
\displaystyle =\left(\frac{1}{3}+\frac{3}{7}\right)x^2y =\left(\frac{7+9}{21}\right)x^2y=\frac{16}{21}x^2y.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Add the following expressions:}
\displaystyle \text{(i) }3x+8y-5z,\ 7x-9y+6z,\ x-2y-3z,\ -5x+3y+z
\displaystyle \text{(ii) }y^2-x^2,\ 2x^2-3y^2,\ 5y^2-3x^2,\ 6x^2+2y^2
\displaystyle \text{(iii) }2p+3r+4q,\ 7r+3p-2q,\ q-r-p,\ 5p+4q-8r
\displaystyle \text{(iv) }7xy+yz-3zx,\ 2yz+zx-5xy,\ 2zx-3yz+4xy
\displaystyle \text{(v) }1-x-x^2-3x^3,\ 2x^2+x^3+3,\ x^2+5x-2,
\displaystyle x^3-x^2-3x
\displaystyle \text{(vi) }4z^4-7z^3+2z-5z^2+3,\ 3z^2+4z^3-2z^4-z-5,
\displaystyle 3z^3-z-1+z^2
\displaystyle \text{(vii) }3+5y-4y^2+7y^3,\ -7+2y+3y^3,
\displaystyle 5-6y-9y^3+2y^2
\displaystyle \text{Answer:}
\displaystyle \text{(i) Arrange the like terms in the same columns:}
\displaystyle \begin{array}{rrrr} 3x&+8y&-5z\\ 7x&-9y&+6z\\ x&-2y&-3z\\ (+)~-5x&+3y&+z\\ \hline 6x&0y&-z \end{array}
\displaystyle \therefore \text{the sum is }6x-z.

\displaystyle \text{(ii) Arrange the like terms in the same columns:}
\displaystyle \begin{array}{rr} -x^2&+y^2\\ 2x^2&-3y^2\\ -3x^2&+5y^2\\ (+)~6x^2&+2y^2\\ \hline 4x^2&+5y^2 \end{array}
\displaystyle \therefore \text{the sum is }4x^2+5y^2.

\displaystyle \text{(iii) Arrange the like terms in the same columns:}
\displaystyle \begin{array}{rrr} 2p&+4q&+3r\\ 3p&-2q&+7r\\ -p&+q&-r\\ (+)~5p&+4q&-8r\\ \hline 9p&+7q&+r \end{array}
\displaystyle \therefore \text{the sum is }9p+7q+r.

\displaystyle \text{(iv) Arrange the like terms in the same columns:}
\displaystyle \begin{array}{rrr} 7xy&+yz&-3zx\\ -5xy&+2yz&+zx\\ (+)~4xy&-3yz&+2zx\\ \hline 6xy&+0yz&+0zx \end{array}
\displaystyle \therefore \text{the sum is }6xy.

\displaystyle \text{(v) Arrange the terms in descending powers of }x\text{:}
\displaystyle \begin{array}{rrrr} -3x^3&-x^2&-x&+1\\ x^3&+2x^2&&+3\\ &+x^2&+5x&-2\\ (+)~x^3&-x^2&-3x&\\ \hline -x^3&+x^2&+x&+2 \end{array}
\displaystyle \therefore \text{the sum is }-x^3+x^2+x+2.

\displaystyle \text{(vi) Arrange the terms in descending powers of }z\text{:}
\displaystyle \begin{array}{rrrrr} 4z^4&-7z^3&-5z^2&+2z&+3\\ -2z^4&+4z^3&+3z^2&-z&-5\\ (+)~&+3z^3&+z^2&-z&-1\\ \hline 2z^4&0z^3&-z^2&0z&-3 \end{array}
\displaystyle \therefore \text{the sum is }2z^4-z^2-3.

\displaystyle \text{(vii) Arrange the terms in descending powers of }y\text{:}
\displaystyle \begin{array}{rrrr} 7y^3&-4y^2&+5y&+3\\ 3y^3&&+2y&-7\\ (+)~-9y^3&+2y^2&-6y&+5\\ \hline y^3&-2y^2&+y&+1 \end{array}
\displaystyle \therefore \text{the sum is }y^3-2y^2+y+1.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Subtract:}
\displaystyle \text{(i) }3y-5z\text{ from }6z-4y
\displaystyle \text{(ii) }3a-5b-2c\text{ from }2a-3b-c
\displaystyle \text{(iii) }p^2-q^2+5r^2\text{ from }q^2-p^2-2r^2
\displaystyle \text{(iv) }5x^2+x-11\text{ from }x^2-2x+8
\displaystyle \text{(v) }1-x+2x^2-x^3\text{ from }2x^3-x^2+3x+2
\displaystyle \text{(vi) }7x^3-6x^2y+9xy^2-2y^3\text{ from}
\displaystyle 3x^2y-2x^3+y^3-5xy^2
\displaystyle \text{Answer:}

\displaystyle \text{(i) Change the signs of the polynomial being subtracted and add:}
\displaystyle \begin{array}{rr} -4y&+6z\\ (+)~-3y&+5z\\ \hline -7y&+11z \end{array}
\displaystyle \therefore \text{the difference is }11z-7y.

\displaystyle \text{(ii) Change the signs of the polynomial being subtracted and add:}
\displaystyle \begin{array}{rrr} 2a&-3b&-c\\ (+)~-3a&+5b&+2c\\ \hline -a&+2b&+c \end{array}
\displaystyle \therefore \text{the difference is }-a+2b+c.

\displaystyle \text{(iii) Change the signs of the polynomial being subtracted and add:}
\displaystyle \begin{array}{rrr} -p^2&+q^2&-2r^2\\ (+)~-p^2&+q^2&-5r^2\\ \hline -2p^2&+2q^2&-7r^2 \end{array}
\displaystyle \therefore \text{the difference is }-2p^2+2q^2-7r^2.

\displaystyle \text{(iv) Change the signs of the polynomial being subtracted and add:}
\displaystyle \begin{array}{rrr} x^2&-2x&+8\\ (+)~-5x^2&-x&+11\\ \hline -4x^2&-3x&+19 \end{array}
\displaystyle \therefore \text{the difference is }-4x^2-3x+19.

\displaystyle \text{(v) Change the signs of the polynomial being subtracted and add:}
\displaystyle \begin{array}{rrrr} 2x^3&-x^2&+3x&+2\\ (+)~x^3&-2x^2&+x&-1\\ \hline 3x^3&-3x^2&+4x&+1 \end{array}
\displaystyle \therefore \text{the difference is }3x^3-3x^2+4x+1.

\displaystyle \text{(vi) Arrange the like terms in the same columns and subtract:}
\displaystyle \begin{array}{rrrr} -2x^3&+3x^2y&-5xy^2&+y^3\\ (+)~-7x^3&+6x^2y&-9xy^2&+2y^3\\ \hline -9x^3&+9x^2y&-14xy^2&+3y^3 \end{array}
\displaystyle \therefore \text{the difference is } -9x^3+9x^2y-14xy^2+3y^3.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Subtract the sum of }4x^2+7xy+3y^2+1
\displaystyle \text{and }2x^2-5xy-2y^2+8\text{ from }9x^2-8xy+11y^2.
\displaystyle \text{Answer:}
\displaystyle \text{First, add the two polynomials:}
\displaystyle \begin{array}{rrrr} 4x^2&+7xy&+3y^2&+1\\ (+)~2x^2&-5xy&-2y^2&+8\\ \hline 6x^2&+2xy&+y^2&+9 \end{array}
\displaystyle \text{Now subtract }6x^2+2xy+y^2+9 \text{ from }9x^2-8xy+11y^2.
\displaystyle \begin{array}{rrrr} 9x^2&-8xy&+11y^2&+0\\ (+)~-6x^2&-2xy&-y^2&-9\\ \hline 3x^2&-10xy&+10y^2&-9 \end{array}
\displaystyle \therefore \text{the required difference is } 3x^2-10xy+10y^2-9.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{What must be added to }2x^2+6x-5\text{ to get}
\displaystyle 3x^2-2x+6?
\displaystyle \text{Answer:}
\displaystyle \text{Required polynomial}=(3x^2-2x+6)-(2x^2+6x-5).
\displaystyle \begin{array}{rrr} 3x^2&-2x&+6\\ (+)~-2x^2&-6x&+5\\ \hline x^2&-8x&+11 \end{array}
\displaystyle \therefore \text{the required polynomial is }x^2-8x+11.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{What must be added to }4-3x+9x^2\text{ to get}
\displaystyle 3-4x^2-x^3?
\displaystyle \text{Answer:}
\displaystyle \text{Required polynomial}=(3-4x^2-x^3)-(4-3x+9x^2).
\displaystyle \begin{array}{rrrr} -x^3&-4x^2&&+3\\ &-9x^2&+3x&-4\\ \hline -x^3&-13x^2&+3x&-1 \end{array}
\displaystyle \therefore \text{the required polynomial is }-x^3-13x^2+3x-1.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{What must be subtracted from }3x^2y-2xy^2+7x-2y
\displaystyle \text{to get }7xy^2-5x^2y-3x+3y?
\displaystyle \text{Answer:}
\displaystyle \text{Required polynomial}=(3x^2y-2xy^2+7x-2y)
\displaystyle -(7xy^2-5x^2y-3x+3y).
\displaystyle \begin{array}{rrrr} 3x^2y&-2xy^2&+7x&-2y\\ (+)~5x^2y&-7xy^2&+3x&-3y\\ \hline 8x^2y&-9xy^2&+10x&-5y \end{array}
\displaystyle \therefore \text{the required polynomial is } 8x^2y-9xy^2+10x-5y.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{What must be subtracted from }a^2+b^2+c^2-3abc
\displaystyle \text{to get }2a^2-b^2-3c^2+abc?
\displaystyle \text{Answer:}
\displaystyle \text{Required polynomial}=(a^2+b^2+c^2-3abc)
\displaystyle -(2a^2-b^2-3c^2+abc).
\displaystyle \begin{array}{rrrr} a^2&+b^2&+c^2&-3abc\\ (+)~-2a^2&+b^2&+3c^2&-abc\\ \hline -a^2&+2b^2&+4c^2&-4abc \end{array}
\displaystyle \therefore \text{the required polynomial is } -a^2+2b^2+4c^2-4abc.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Subtract }a^3-3a^2\text{ from }b^3-3b^2.
\displaystyle \text{Answer:}
\displaystyle (b^3-3b^2)-(a^3-3a^2) =b^3-3b^2-a^3+3a^2.
\displaystyle \therefore \text{the difference is }b^3-3b^2-a^3+3a^2.
\displaystyle \\

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