Given:The given equations are:(i) $9\frac{1}{4}=y-1\frac{1}{3}$(ii) $\frac{5x}{3}+\frac{2}{5}=1$To do:We have to solve the given equations and verify the solutions.Solution:To verify the solutions we have to find the values of the variables and substitute them in the equation. Find the value of LHS and the value of RHS and check whether both are equal.(i) The given equation is $9\frac{1}{4}=y-1\frac{1}{3}$.$9\frac{1}{4}=y-1\frac{1}{3}$$\frac{9\times4+1}{4}=y-\frac{1\times3+1}{3}$$\frac{36+1}{4}=y-\frac{3+1}{3}$$\frac{37}{4}=y-\frac{4}{3}$$y=\frac{37}{4}+\frac{4}{3}$                   (Transposing $\frac{4}{3}$)LCM of the denominators $4$ and $3$ is $12$.$y=\frac{37}{4}+\frac{4}{3}$$y=\frac{37\times3+4\times4}{12}$$y=\frac{111+16}{12}$$y=\frac{127}{12}$Verification:LHS $=9\frac{1}{4}$$=\frac{9\times4+1}{4}$$=\frac{36+1}{4}$$=\frac{37}{4}$RHS $=y-1\frac{1}{3}$$=\frac{127}{12}-1\frac{1}{3}$$=\frac{127}{12}-\frac{1\times3+1}{3}$$=\frac{127}{12}-\frac{3+1}{3}$$=\frac{127}{12}-\frac{4}{3}$$=\frac{127-4\times4}{12}$$=\frac{127-16}{12}$$=\frac{111}{12}$$=\frac{3\times37}{3\times4}$$=\frac{37}{4}$LHS $=$ RHSHence verified.(ii) The given equation is $\frac{5x}{3}+\frac{2}{5}=1$.$\frac{5x}{3}+\frac{2}{5}=1$LCM of denominators $3$ and $5$ is $15$$\frac{5x \times 5+2\times3}{15}=1$$\frac{25x+6}{15}=1$On cross multiplication, we get, $25x+6=15$$25x=15-6$$25x=9$$x=\frac{9}{25}$Verification:LHS $=\frac{5x}{3}+\frac{2}{5}$$=\frac{5\times \frac{9}{25}}{3}+\frac{2}{5}$$=\frac{1\times \frac{3}{5}}{1}+\frac{2}{5}$$=\frac{3}{5}+\frac{2}{5}$$=\frac{3+2}{5}$$=\frac{5}{5}$$=1$RHS $=1$LHS $=$ ... Read More

Given:The given expressions are:(i) $acx^2+(bc+ad)x+bd$ by $ax+b$(ii) $(a^2+2ab+b^2)-(a^2+2ac+c^2)$ by $2a+b+c$To do:We have to divide the given expressions.Solution:We have to divide the given polynomials by simplifying them using algebraic formulas.Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.Therefore, (i) The given expression is $acx^2+(bc+ad)x+bd$ by $ax+b$.$acx^2+(bc+ad)x+bd \div (ax+b)=\frac{acx^2+(bc+ad)x+bd}{ax+b}$$acx^2+(bc+ad)x+bd \div (ax+b)=\frac{acx^2+bcx+adx+bd}{ax+b}$$acx^2+(bc+ad)x+bd \div (ax+b)=\frac{cx(ax+b)+d(ax+b)}{ax+b}$        (Taking $cx$ and $d$ common)$acx^2+(bc+ad)x+bd \div (ax+b)=\frac{cx(ax+b)}{ax+b}+\frac{d(ax+b)}{ax+b}$$acx^2+(bc+ad)x+bd \div (ax+b)=cx+d$Hence, $acx^2+(bc+ad)x+bd$ divided by $ax+b$ is $cx+d$.(ii) The given expression is $(a^2+2ab+b^2)-(a^2+2ac+c^2)$ by $2a+b+c$.$(a^2+2ab+b^2)-(a^2+2ac+c^2) \div (2a+b+c)=\frac{(a^2+2ab+b^2)-(a^2+2ac+c^2)}{2a+b+c}$$(a^2+2ab+b^2)-(a^2+2ac+c^2) \div (2a+b+c)=\frac{(a+b)^2-(a+c)^2}{2a+b+c}$                [Since $(x+y)^2=x^2+2xy+y^2$]$(a^2+2ab+b^2)-(a^2+2ac+c^2) \div (2a+b+c)=\frac{(a+b+a+c)(a+b-a-c)}{2a+b+c}$      ... Read More

Given:The given expressions are:(i) $ax^2-ay^2$ by $ax+ay$(ii) $x^4-y^4$ by $x^2-y^2$To do:We have to divide the given expressions.Solution:We have to divide the given polynomials by simplifying them using algebraic formulas.Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.Therefore, (i) The given expression is $ax^2-ay^2$ by $ax+ay$.$ax^2-ay^2$ can be written as, $ax^2-ay^2=a(x^2-y^2)$            (Taking $a$ common)$ax^2-ay^2=a(x+y)(x-y)$.........(I)               [Since $a^2-b^2=(a+b)(a-b)$]Therefore, $ax^2-ay^2 \div (ax+ay)=\frac{ax^2-ay^2}{ax+ay}$$ax^2-ay^2 \div (ax+ay)=\frac{a(x+y)(x-y)}{a(x+y)}$             [Using (I) and taking $a$ common in $ax+ay$]$ax^2-ay^2 \div (ax+ay)=(x-y)$Hence, $ax^2-ay^2$ divided by $ax+ay$ is ... Read More

Given:The given expressions are:(i) $5x^3-15x^2+25x$ by $5x$(ii) $4z^3+6z^2-z$ by $\frac{-1}{2}z$(iii) $9x^2y-6xy+12xy^2$ by $\frac{-3}{2}xy$To do:We have to divide the given expressions.Solution:We have to divide the given polynomials by monomials using the formula $x^a \div x^b=a^{a-b}$Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.Monomial:A monomial is an expression that contains a single term composed of a product of constants and variables with non-negative integer exponents. Therefore, (i) The given expression is $5x^3-15x^2+25x$ by $5x$.$5x^3-15x^2+25x \div 5x=\frac{5x^3}{5x}-\frac{15x^2}{5x}+\frac{25x}{5x}$$5x^3-15x^2+25x \div 5x=\frac{5}{5}x^{3-1}-\frac{15}{5}x^{2-1}+\frac{25}{5}x^{1-1}$$5x^3-15x^2+25x \div 5x=x^{2}-3x^{1}+5x^{0}$$5x^3-15x^2+25x \div 5x=x^{2}-3x+5$             [Since $x^0=1$]Hence, $5x^3-15x^2+25x$ divided by $5x$ is ... Read More

Given:The given expressions are:(i) $-x^6+2x^4+4x^3+2x^2$ by $\sqrt2x^2$(ii) $-4a^3+4a^2+a$ by $2a$(iii) $\sqrt3a^4+2\sqrt3a^3+3a^2-6a$ by $3a$To do:We have to divide the given expressions.Solution:We have to divide the given polynomials by monomials using the formula $x^a \div x^b=a^{a-b}$Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.Monomial:A monomial is an expression that contains a single term composed of a product of constants and variables with non-negative integer exponents. Therefore, (i) The given expression is $-x^6+2x^4+4x^3+2x^2$ by $\sqrt2x^2$.$-x^6+2x^4+4x^3+2x^2 \div \sqrt2x^2=\frac{-x^6}{\sqrt2x^2}+\frac{2x^4}{\sqrt2x^2}+\frac{4x^3}{\sqrt2x^2}+\frac{2x^2}{\sqrt2x^2}$$-x^6+2x^4+4x^3+2x^2 \div \sqrt2x^2=\frac{-1}{\sqrt2}x^{6-2}+\frac{\sqrt2 \times \sqrt2}{\sqrt2}x^{4-2}+\frac{2\sqrt2 \times \sqrt2}{\sqrt2}x^{3-2}+\frac{\sqrt2 \times \sqrt2}{\sqrt2}x^{2-2}$$-x^6+2x^4+4x^3+2x^2 \div \sqrt2x^2=\frac{-1}{\sqrt2}x^{4}+\frac{\sqrt2}{1}x^{2}+\frac{2\sqrt2}{1}x^{1}+\frac{\sqrt2}{1}x^{0}$$-x^6+2x^4+4x^3+2x^2 \div \sqrt2x^2=\frac{-1}{\sqrt2}x^{4}+\sqrt2x^{2}+2\sqrt2x+\sqrt2$             [Since ... Read More

Given:The given expressions are:(i) $x+2x^2+3x^4-x^5$ by $2x$(ii) $y^4-3y^3+\frac{1}{2y^2}$ by $3y$(iii) $-4a^3+4a^2+a$ by $2a$To do:We have to divide the given expressions.Solution:We have to divide the given polynomials by monomials using the formula $x^a \div x^b=a^{a-b}$Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.Monomial:A monomial is an expression that contains a single term composed of a product of constants and variables with non-negative integer exponents. Therefore, (i) The given expression is $x+2x^2+3x^4-x^5$ by $2x$.$x+2x^2+3x^4-x^5 \div 2x=\frac{x}{2x}+\frac{2x^2}{2x}+\frac{3x^4}{2x}-\frac{x^5}{2x}$$x+2x^2+3x^4-x^5 \div 2x=\frac{1}{2}+x^{2-1}+\frac{3}{2}x^{4-1}-\frac{1}{2}x^{5-1}$$x+2x^2+3x^4-x^5 \div 2x=\frac{1}{2}+x^{1}+\frac{3}{2}x^{3}-\frac{1}{2}x^{4}$$x+2x^2+3x^4-x^5 \div 2x=\frac{1}{2}+x+\frac{3}{2}x^{3}-\frac{1}{2}x^{4}$Hence, $x+2x^2+3x^4-x^5$ divided by $2x$ is $\frac{1}{2}+x+\frac{3}{2}x^{3}-\frac{1}{2}x^{4}$.(ii) The given expression is $y^4-3y^3+\frac{1}{2y^2}$ by $3y$.$y^4-3y^3+\frac{1}{2y^2} \div ... Read More

Given:The given expressions are:(i) $\frac{16m^3y^2}{4m^2y}$(ii) $\frac{32m^2n^3p^2}{4mnp}$To do:We have to simplify the given expressions.Solution:We have to divide the given polynomials by monomials using the formula $x^a \div x^b=a^{a-b}$Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.Monomial:A monomial is an expression that contains a single term composed of a product of constants and variables with non-negative integer exponents. Therefore, (i) The given expression is $\frac{16m^3y^2}{4m^2y}$$\frac{16m^3y^2}{4m^2y}=\frac{16}{4}m^{3-2}y^{2-1}$$\frac{16m^3y^2}{4m^2y}=4m^{1}y^{1}$$\frac{16m^3y^2}{4m^2y}=4my$Hence,  $\frac{16m^3y^2}{4m^2y}=4my$.(ii) The given expression is $\frac{32m^2n^3p^2}{4mnp}$.$\frac{32m^2n^3p^2}{4mnp}=\frac{32}{4}m^{2-1}n^{3-1}p^{2-1}$$\frac{32m^2n^3p^2}{4mnp}=8m^{1}n^{2}p^{1}$$\frac{32m^2n^3p^2}{4mnp}=8mn^2p$Hence, $\frac{32m^2n^3p^2}{4mnp}=8mn^2p$.Read More

Given:The given expressions are:(i) $ -21abc^2$ by $7abc$(ii) $72xyz^2$ by $-9xz$(iii) $-72a^4b^5c^8$ by $-9a^2b^2c^3$To do:We have to divide the given expressions.Solution:We have to divide the given polynomials by monomials using the formula $x^a \div x^b=a^{a-b}$Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.Monomial:A monomial is an expression that contains a single term composed of a product of constants and variables with non-negative integer exponents. Therefore, (i) The given expression is $-21abc^2$ by $7abc$.$-21abc^2 \div 7abc=\frac{-21}{7}a^{1-1}b^{1-1}c^{2-1}$$-21abc^2 \div 7abc=-3a^{0}b^{0}c^{1}$$-21abc^2 \div 7abc=-3c$                    [Since $m^0=1$]Hence, $-21abc^2$ divided ... Read More

Given:The given expressions are:(i) $6x^3y^2z^2$ by $3x^2yz$(ii) $15m^2n^3$ by $5m^2n^2$(iii) $24a^3b^3$ by $-8ab$To do:We have to divide the given expressions.Solution:We have to divide the given polynomials by monomials using the formula $x^a \div x^b=a^{a-b}$Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.Monomial:A monomial is an expression that contains a single term composed of a product of constants and variables with non-negative integer exponents. Therefore, (i) The given expression is $6x^3y^2z^2$ by $3x^2yz$.$6x^3y^2z^2 \div 3x^2yz=\frac{6}{3}x^{3-2}y^{2-1}z^{2-1}$$6x^3y^2z^2 \div 3x^2yz=2x^{1}y^{1}z^{1}$$6x^3y^2z^2 \div 3x^2yz=2xyz$Hence,  $6x^3y^2z^2$ divided by $3x^2yz$ is $2xyz$.(ii) The given expression is $15m^2n^3$ by $5m^2n^2$.$15m^2n^3 \div 5m^2n^2=\frac{15}{5}m^{2-2}n^{3-2}$$15m^2n^3 \div 5m^2n^2=3m^{0}n^{1}$$15m^2n^3 ... Read More

Given:The given expressions are:(i) $x^2+3+6x+5x^4$(ii) $a^2+4+5a^6$(iii) $(x^3-1)(x^3-4)$(iv) $(a^3-\frac{3}{8})(a^3+\frac{16}{17})$(v) $(a+\frac{3}{4})(a+\frac{4}{3})$To do:We have to write the standard form of the given polynomials and find their degree.Solution:Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.The standard form of the polynomial is a polynomial where the terms are arranged in descending order by degree. Degree of a polynomial:The degree of a polynomial is the highest or the greatest power of a variable in the polynomial expression.To find the degree, identify the exponents on the variables in each term, and add them together to find the ... Read More