Multiplication and Division of Algebraic Expressions
Category : 8th Class
Various Algebraic Relations to be used in this chapter
1. \[{{(A+B)}^{2}}={{A}^{2}}+{{B}^{2}}+2AB\]
2. \[{{(A-B)}^{2}}={{A}^{2}}+{{B}^{2}}-2AB\]
3. \[{{A}^{2}}-{{B}^{2}}=(A-B)(A+B)\]
4. \[{{(A+B+C)}^{2}}={{A}^{2}}+{{B}^{2}}+{{C}^{2}}+2AB+2BC+2CA\]
5. \[{{A}^{3}}+{{B}^{3}}+{{C}^{3}}-3ABC=(A+B+C)({{A}^{2}}+{{B}^{2}}+{{C}^{2}}\]\[-AB-BC-CA)\]
6. \[{{(A+B)}^{3}}={{A}^{3}}+{{B}^{3}}+3{{A}^{2}}B+3A{{B}^{2}}\]
7. \[{{(A-B)}^{3}}={{A}^{3}}-{{B}^{3}}-3{{A}^{2}}B+3A{{B}^{2}}\]
8. \[{{(A+B)}^{4}}={{A}^{4}}+4{{A}^{3}}B+6{{A}^{2}}{{B}^{2}}+4A{{B}^{3}}+{{B}^{4}}\]
9. \[{{(A-B)}^{4}}={{A}^{4}}-4{{A}^{3}}B+6{{A}^{2}}{{B}^{2}}-4A{{B}^{3}}+{{B}^{4}}\]
10. \[{{A}^{3}}-{{B}^{3}}=(A+B)({{A}^{2}}+{{B}^{2}}-AB)\]
11. \[{{A}^{3}}-{{B}^{3}}=(A-B)({{A}^{2}}+{{B}^{2}}+AB)\]
Find the product of \[(2{{x}^{2}}-5x+4)\] and \[({{x}^{2}}+7x-8)\]
(a) \[(2{{x}^{4}}-9{{x}^{3}}-47{{x}^{2}}+68x+32)\]
(b) \[(2{{x}^{4}}+9{{x}^{3}}-47{{x}^{2}}+68x-32)\]
(c) \[(2{{x}^{4}}-9{{x}^{3}}-47{{x}^{2}}+68x+32)\]
(d) \[(2{{x}^{4}}-9{{x}^{3}}-47{{x}^{2}}-68x-32)\]
(e) None of these
Answer: (b)
The product of \[(3x+5y)\] and \[(5x-7y)\] is.
(a) \[15{{x}^{2}}+4xy-35{{y}^{2}}\]
(b) \[5{{x}^{2}}-4xy+35{{y}^{2}}\]
(c) \[5{{x}^{5}}+4xy+35{{y}^{2}}\]
(d) \[{{x}^{2}}-4xy-35y\]
(e) None of these
Answer: (a)
Explanation:
\[=(3x+5y)(5x-7y)\]
\[=3x(5x-7y)+5y(5x-7y)\]
\[=15{{x}^{2}}-21xy+25xy-35{{y}^{2}}\]
\[=15{{x}^{2}}+4xy-35{{y}^{2}}\]
The product of \[(-3{{x}^{2}}y)(4{{x}^{2}}y-3x{{y}^{2}}+4x-5y)\] is .........
(a) \[12{{x}^{3}}{{y}^{2}}+9{{x}^{2}}{{y}^{2}}-12{{x}^{4}}y+15{{x}^{3}}{{y}^{3}}\]
(b) \[-12{{x}^{4}}{{y}^{2}}+9{{x}^{3}}{{y}^{3}}-12{{x}^{3}}y+15{{x}^{2}}{{y}^{2}}\]
(c) \[-12x{{y}^{2}}+9{{x}^{3}}{{y}^{2}}-12{{x}^{3}}y+15xy\]
(d) \[-12x{{y}^{3}}+9xy-12x{{y}^{2}}+15{{x}^{2}}y\]
(e) None of these
Answer: (b)
Explanation:
\[(-3{{x}^{2}}y)(4{{x}^{2}}y-3x{{y}^{2}}+4x-5y)=\]\[-12{{x}^{4}}{{y}^{2}}+9{{x}^{3}}{{y}^{3}}-12{{x}^{4}}{{y}^{2}}+9{{x}^{3}}{{y}^{3}}-12{{x}^{3}}y+15{{x}^{2}}{{y}^{2}}\]
Therefore, option (b) is correct and rest of options is incorrect.
The product of \[(3{{x}^{2}}+{{y}^{2}})\] and \[(2{{x}^{2}}+3{{y}^{2}})\] is ------.
(a) \[{{x}^{3}}+10{{x}^{2}}{{y}^{3}}+3{{y}^{4}}\]
(b) \[3{{x}^{2}}+5{{x}^{5}}{{y}^{2}}+3{{y}^{4}}\]
(c) \[6{{x}^{5}}+10{{x}^{2}}{{y}^{3}}+3{{y}^{4}}\]
(d)\[6{{x}^{4}}+11{{x}^{2}}{{y}^{2}}+30{{y}^{4}}\]
(e) None of these
Answer: (d)
Explanation:
\[(3{{x}^{2}}+{{y}^{2}})(2{{x}^{2}}+3{{y}^{2}})\]
\[=6{{x}^{4}}+9{{x}^{2}}{{y}^{2}}+2{{x}^{2}}{{y}^{2}}+30{{y}^{4}}\]
\[=6{{x}^{4}}+11{{x}^{2}}{{y}^{2}}+30{{y}^{4}}\]
Multiply: \[(3x+2y)(3x+2y)\]
(a) \[9{{x}^{2}}+4{{y}^{2}}+12xy\]
(b) \[18{{x}^{3}}+2{{y}^{2}}+10xy\]
(c) \[9{{x}^{2}}+4{{y}^{2}}+8xy\]
(d) \[9{{x}^{2}}+6{{y}^{3}}+12xy\]
(e) None of these
Answer: (a)
Explanation:
\[(3x+2y)+(3x+2y)=3x(3x+2y)+2y(3xx+2y)\]
\[9{{x}^{2}}+6xy+6xy+4{{y}^{2}}=9{{x}^{2}}+12xy+4{{y}^{2}}\]
Solve: \[(4{{x}^{2}}+5)(4{{x}^{2}}+5)\]
(a) \[16{{x}^{4}}+25+40{{x}^{2}}\]
(b) \[16{{x}^{4}}+28+30{{x}^{2}}\]
(c) \[16{{x}^{4}}+30+20{{x}^{2}}\]
(d) \[16{{x}^{4}}+8+25{{x}^{2}}\]
(e) None of these
Answer: (a)
Explanation:
\[(4{{x}^{2}}+5)(4{{x}^{2}}+5)={{(4{{x}^{2}}+5)}^{2}}\]
\[{{(4{{x}^{2}})}^{2}}={{5}^{2}}+2(4{{x}^{2}})\times 5\] [Using\[~{{(\text{a}+\text{b})}^{\text{2}}}={{\text{a}}^{\text{2}}}+{{\text{b}}^{\text{2}}}+\text{2ab}\]]
\[=16{{x}^{4}}+25+40{{x}^{2}}\]
Multiplication of both the expressions is as same as option (a).
\[(4x-7y)(4x-7y)\] equal to:
(a) \[16{{x}^{2}}+56xy+49{{y}^{2}}\]
(b) \[16{{x}^{2}}-8xy+49{{y}^{2}}\]
(c) \[16{{x}^{2}}-56xy-49{{y}^{2}}\]
(d)\[16{{x}^{2}}-56xy+49{{y}^{2}}\]
(e) None of these
Answer: (d)
Explanation:
\[(4x-7y)(4x-7y)=(4{{x}^{2}})-2.4x.7y+{{(7y)}^{2}}=\]\[16{{x}^{2}}=56xy+49{{y}^{2}}\]
Solve the expression: \[\left( x-\frac{3}{x} \right)\left( x-\frac{3}{x} \right)=\]
(a) \[{{x}^{2}}+\frac{9}{{{x}^{2}}}-18\]
(b) \[{{x}^{3}}+\frac{9}{{{x}^{2}}}-6\]
(c) \[{{x}^{2}}+\frac{9}{{{x}^{2}}}-6\]
(d) \[{{x}^{2}}+\frac{1}{{{x}^{2}}}-6\]
(e) None of these
Answer: (c)
Explanation:
\[\left( x-\frac{3}{x} \right)\left( x-\frac{3}{x} \right)={{x}^{2}}{{\left( \frac{3}{x} \right)}^{2}}-2\times x\times \frac{3}{x}={{x}^{2}}+\frac{9}{{{x}^{2}}}-6\]
Simplify: \[(4x+5y)(4x-5y)\]
(a) \[16{{x}^{2}}-25xy\]
(b) \[16{{x}^{2}}-5{{y}^{2}}\]
(c) \[16{{x}^{2}}-25{{y}^{2}}\]
(d) \[16{{x}^{2}}-{{y}^{2}}\]
(e) None of these
Answer: (c)
Explanation:
\[(4x+5y)(4x-5y)={{(4x)}^{2}}-{{(-5y)}^{2}}=16{{x}^{2}}-25{{y}^{2}}\]
Simplify: \[(2x+3y)(2x-3y)\]
(a) \[{{x}^{2}}-3{{y}^{2}}\]
(b) \[2{{x}^{2}}-3{{y}^{2}}\]
(c) \[4{{x}^{2}}-9{{y}^{2}}\]
(d) \[{{x}^{2}}-9{{y}^{2}}\]
(e) None of these
Answer: (c)
Explanation:
\[(2x+3y)(2x-3y)={{(2x)}^{2}}-{{(3y)}^{2}}=4{{x}^{2}}-9{{y}^{2}}\]
Find the sum of the given algebraic expression. \[2{{p}^{2}}{{q}^{2}}-3pq+4,4+7{{p}^{2}}{{q}^{2}}-8pq,\]\[5{{p}^{2}}{{q}^{2}}-8+9pq\]
(a) \[1{{p}^{2}}{{q}^{2}}-5pq+5\]
(b) \[14{{p}^{2}}{{q}^{2}}-2pq+1\]
(c) \[11{{p}^{2}}{{q}^{2}}+5pq+10\]
(d)\[14{{p}^{2}}{{q}^{2}}+19pq+17\]
(e) None of these
Answer: (b)
Find the sum of the given expressions: \[-5{{x}^{2}}+3x-8,\] \[4x+7x-2{{x}^{2}}\] and \[6-2x+2{{x}^{2}}\]
(a) 2x+3y+z
(b) x+2y+3z
(c) 9x+6y+4z
(d) -5x2 + 5x + 5
(e) None of these
Answer: (d)
Explanation:
Writing the given expression in descending powers of\[x\]in the form of rows with the like terms as shown below and adding them column wise we get,
\[\begin{align} & -5{{x}^{2}}+3x-8 \\ & -2{{x}^{2}}+4x+7 \\ & \,\,\,\underline{2{{x}^{2}}-2x+6} \\ & -\underline{5{{x}^{2}}+5x+5} \\ \end{align}\]
Subtract: \[\text{3a}-\text{3b}+\text{c}\] from \[\text{4a}+\text{5b}-\text{3c}\]
(a) \[\text{2a}+\text{3b}+\text{c}\]
(b) \[\text{a}+\text{2b}+\text{3c}\]
(c) \[\text{a}+\text{8b}-\text{4c}\]
(d)\[-5{{x}^{2}}+5x+5\]
(e) None of these
Answer: (c)
Explanation:
\[\begin{align} & \,\,\,\,\,\text{4a}+\text{5b}-\text{3c} \\ & \pm \,\underline{\text{3a }+\text{ 3b }\pm \text{ c}} \\ & \,\,\,\,\underline{\text{a }+\text{ 8b}-\text{4c}} \\ \end{align}\]
What must be subtracted form \[3{{a}^{2}}-6ab-3{{b}^{2}}-1\] to get \[4{{a}^{2}}-7ab-4{{b}^{2}}+1\]?
(a) \[-{{a}^{2}}+ab+{{b}^{2}}-2\]
(b) \[{{a}^{2}}+ab+{{b}^{2}}+2\]
(c) \[{{a}^{2}}-ab-{{b}^{2}}+2\]
(d) \[{{a}^{2}}-ab-4{{b}^{2}}-2\]
(e) None of these
Answer: (a)
Explanation:
\[\begin{align} & \text{3}{{\text{a}}^{\text{2}}}-\text{6ab}-\text{3}{{\text{b}}^{\text{2}}}-1 \\ & \underline{\pm \text{4}{{\text{a}}^{\text{2}}}\mp \text{7ab}\mp 4{{\text{b}}^{\text{2}}}\pm 1} \\ & \,\,\,-{{a}^{2}}+ab+{{b}^{2}}-2 \\ \end{align}\]
Find the product of \[5{{m}^{2}}n,-3mnp\] and \[-5{{n}^{2}}p\]
(a) \[75{{m}^{3}}{{n}^{3}}{{p}^{2}}\]
(b) \[75{{m}^{2}}{{n}^{4}}{{p}^{2}}\]
(c) \[75m{{n}^{3}}{{p}^{3}}\]
(d) \[75{{m}^{3}}n{{p}^{2}}\]
(e) None of these
Answer: (b)
Explanation:
\[5{{m}^{2}}n\times (-3mnp)\times (-5{{n}^{2}}p)=75{{m}^{3}}{{n}^{4}}{{p}^{2}}\]
Simplify \[3{{x}^{2}}{{y}^{2}}(5{{x}^{2}}-4xy+6{{y}^{2}})\]
(a) \[15{{x}^{4}}{{y}^{2}}-12{{x}^{3}}{{y}^{3}}+18{{y}^{4}}{{x}^{2}}\]
(b) \[15{{x}^{2}}{{y}^{2}}-12{{x}^{3}}{{y}^{3}}+18{{y}^{2}}{{x}^{2}}\]
(c) \[15{{x}^{2}}{{y}^{3}}-12{{x}^{2}}{{y}^{2}}+6{{y}^{2}}{{y}^{2}}\]
(d) \[15x{{y}^{3}}-12{{x}^{3}}{{y}^{3}}+18{{x}^{2}}{{y}^{4}}\]
(e) None of these
Answer: (a)
Explanation:
\[3{{x}^{2}}{{y}^{2}}(-5{{x}^{2}}-4xy+6{{y}^{2}})=15{{x}^{4}}{{y}^{2}}-12{{x}^{3}}{{y}^{3}}+18{{y}^{4}}{{x}^{2}}\]
The product of \[(5{{x}^{2}}-6x+9)\] and \[(2x-3)\] is----.
(a) \[10{{x}^{3}}-27{{x}^{2}}+36x-27\]
(b) \[10{{x}^{2}}-26{{x}^{2}}+36x-27\]
(c) \[10{{x}^{3}}-27{{x}^{2}}-36x-27\]
(d) \[10{{x}^{3}}-27{{x}^{2}}+36x+27\]
(e) None of these
Answer: (a)
Explanation:
\[(5{{x}^{2}}-6x+9)(2x-3)\] \[=10{{x}^{3}}-27{{x}^{2}}+36x-27\]
The product of \[(2{{x}^{2}}-5{{x}^{2}}-x+7)\] and \[(3-2x+{{x}^{2}})\] is -------.
(a) \[-3{{x}^{4}}+5{{x}^{3}}-17x+21\]
(b) \[{{x}^{5}}+24{{x}^{4}}+5{{x}^{2}}+x+21\]
(c) \[8{{x}^{5}}+{{x}^{4}}-12+7x+1\]
(d) \[3{{x}^{5}}-4{{x}^{4}}+1{{x}^{3}}-5{{x}^{2}}-7x+2\]
(e) None of these
Answer: (a)
Explanation:
\[=(2{{x}^{2}}-5{{x}^{2}}-x+7)(3-2x+{{x}^{2}})\] \[=6{{x}^{2}}-4{{x}^{3}}+2{{x}^{4}}-15{{x}^{2}}+10{{x}^{3}}-5{{x}^{4}}-3x\]\[2{{x}^{2}}-{{x}^{3}}+21-14x+7{{x}^{2}}\] \[=-3{{x}^{4}}+5{{x}^{3}}-17x+21\]
Find the quotient of polynomial if \[{{y}^{3}}-6{{y}^{2}}+9y-2\] is divided by \[y-2\].
(a) \[{{y}^{2}}+4y+1\]
(b) \[{{y}^{2}}-4y+1\]
(c) \[{{y}^{2}}+4y-1\]
(d) \[{{y}^{2}}-4y-1\]
(e) None of these
Answer: (b)
Explanation:
\[y-2\overset{{{y}^{2}}-4y-1}{\overline{\left){\begin{align} & {{y}^{3}}-6{{y}^{2}}+9y-2 \\ & \underline{\pm {{y}^{3}}\mp 2{{y}^{2}}} \\ & \,\,\,\,\,\,\,\,\,\,\,-4{{y}^{2}}+9y \\ & \,\,\,\,\,\,\,\,\,\,\,\underline{\mp 4{{y}^{2}}\pm 8y} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y-2 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\pm y\mp 2} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \\ \end{align}}\right.}}\]
Find the remainder when \[\text{5}{{\text{m}}^{\text{3}}}-\text{13}{{\text{m}}^{\text{2}}}+\text{15m}+\] 7 is divided by \[\text{4}-\text{3m}+{{\text{m}}^{\text{2}}}\].
(a) m-1
(b) m+1
(c) 2m+1
(d) 2m-1
(e) None of these
Answer: (a)
Explanation:
\[\text{4}-\text{3m}+{{\text{m}}^{\text{2}}}\overset{\text{5m}+\text{2}}{\overline{\left){\begin{align} & \text{5}{{\text{m}}^{\text{3}}}-\text{13}{{\text{m}}^{\text{2}}}+\text{15m}+\text{7} \\ & \underline{\pm \text{5}{{\text{m}}^{\text{3}}}\mp \text{l5}{{\text{m}}^{\text{2}}}\pm \text{2}0\text{m}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2{{m}^{2}}-5m+7 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\mp 2{{m}^{2}}\mp 6m+8} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m-1 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\pm m\pm 1} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \\ \end{align}}\right.}}\]
Divide \[2{{x}^{2}}+3x+1\] by \[(x+1)\] then quotient is:
(a) \[2x+1\]
(b) \[5x+1\]
(c) \[4x+1\]
(d) \[6x+1\]
(e) None of these
Answer: (a)
Explanation:
\[x+1\overset{2x+1}{\overline{\left){\begin{align} & 2{{x}^{2}}+3x+1 \\ & \underline{\pm 2{{x}^{2}}\pm 2x} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x+1 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\pm x\pm 1} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,00 \\ \end{align}}\right.}}\]
Quotient \[=2x+1\]
Remainder = 0
Divide \[(29x-6{{x}^{2}}-28)\] by \[(3x-4)\] and find the quotient
(a) \[2x+7\]
(b) \[-2x+7\]
(c) \[2x-7\]
(d) \[2x+6\]
(e) None of these
Answer: (b)
Explanation:
\[3x-4\overset{-2x+7}{\overline{\left){\begin{align} & -6{{x}^{2}}+29x-28 \\ & \underline{\mp 6{{x}^{2}}\pm 8x} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,21x-28 \\ & \,\,\,\,\,\,\,\,\,\,\,\underline{\pm 21x\mp 28} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,00 \\ \end{align}}\right.}}\]
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