# 8th Class Mathematics Algebraic Expression Multiplication and Division of Algebraic Expressions

## Multiplication and Division of Algebraic Expressions

Category : 8th Class

### Multiplication and Division of Algebraic Expressions

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

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

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

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

Explanation:

$(3{{x}^{2}}+{{y}^{2}})(2{{x}^{2}}+3{{y}^{2}})$

$=6{{x}^{4}}+9{{x}^{2}}{{y}^{2}}+2{{x}^{2}}{{y}^{2}}+30{{y}^{4}}$

$=6{{x}^{4}}+11{{x}^{2}}{{y}^{2}}+30{{y}^{4}}$

#### Problems Based on Identities

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

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

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

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

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

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

Explanation:

$(2x+3y)(2x-3y)={{(2x)}^{2}}-{{(3y)}^{2}}=4{{x}^{2}}-9{{y}^{2}}$

• An expression with one or more terms is called a polynomial.
• It can be used to find the sum of any order of the polynomials.
• A Strange Prime Number - The prime number 73,939,133 has a very strange; property. If you keep removing a digit from the right hand end of the number, each of the remaining numbers is also prime.
• At the rate of moving 1 disc per second around the clock, it is estimated to take about 585 billion years to complete the transfer of the gold discs in the tower of Brahma.

• While adding or subtracting the algebraic expression we always add or subtract; the like terms.
• While multiplying we multiply each term of the expression with each term of the other expression.
• The constants multiplied with the variables are called the coefficient of the term.
• If the variables and powers of the terms are same then they are called like terms otherwise they are called unlike terms.
• An identity is an equality, which is true for all values of the variables in the equality.
• An algebraic expression consists of variables and constants.

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

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

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

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

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

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

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

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

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

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

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

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

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.}}