Given:The given expressions are:(i) $x^2+2x^{-2}$(ii) $\sqrt{ax}+x^2-x^3$(iii) $3y^3-\sqrt{5}y+9$(iv) $ax^{\frac{1}{2}}y^7+ax+9x^2+4$(v) $3x^{-3}+2x^{-1}+4x+5$To do:We have to find which of the given expressions are polynomials.Solution:Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.To identify whether the given expression is polynomial, check if all the powers of the variables are whole numbers after simplification. If any of the powers is a fraction or negative integer then it is not a polynomial.(i) The given expression is $x^2+2x^{-2}$.The term $2x^{-2}$ has a negative power of $-2$.Therefore, The given expression is not a polynomial.(ii) The given expression is $\sqrt{ax}+x^2-x^3$.The term $\sqrt{ax}=\sqrt{a}x^{\frac{1}{2}}$ has ... Read More

Given:The given polynomials are:(i) $2x^3+5x^2-7$(ii) $5x^2-3x+2$(iii) $2x+x^2-8$(iv) $\frac{1}{2}y^7-12y^6+48y^5-10$(v) $3x^3+1$(vi) $5$(vii) $20x^3+12x^2y^2-10y^2+20$To do:We have to find the degree of each of the given polynomials.Solution: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 degree of each term.(i) The given polynomial is $2x^3+5x^2-7$The variable in the given polynomial is $x$.Here, The power of $x$ in $2x^3$ is $3$.Therefore, The degree of the given polynomial is $3$.(ii) The given polynomial is $5x^2-3x+2$The variable in ... Read More

Given:The given quadratic polynomials are:(i) $y^2-7y+12$(ii) $z^2-4z-12$To do:We have to factorize the given quadratic polynomials.Solution:Factorizing algebraic expressions:Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. An algebraic expression is factored completely when it is written as a product of prime factors.Completing the square is a method that is used to write a quadratic expression in a way such that it contains the perfect square.(i) The given expression is $y^2-7y+12$.Here, The coefficient of $y^2$ is $1$The coefficient of $y$ is $-7$The constant term is $12$Coefficient of $y^2$ is $1$. So, we ... Read More

Given:The given quadratic polynomials are:(i) $a^2+2a-3$(ii) $4x^2-12x+5$To do:We have to factorize the given quadratic polynomials.Solution:Factorizing algebraic expressions:Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. An algebraic expression is factored completely when it is written as a product of prime factors.Completing the square is a method that is used to write a quadratic expression in a way such that it contains the perfect square.(i) The given expression is $a^2+2a-3$.Here, The coefficient of $a^2$ is $1$The coefficient of $a$ is $2$The constant term is $-3$Coefficient of $a^2$ is $1$. So, we ... Read More

Given:The given quadratic polynomials are:(i) $x^2+12x+20$(ii) $a^2-14a-51$To do:We have to factorize the given quadratic polynomials.Solution:Factorizing algebraic expressions:Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. An algebraic expression is factored completely when it is written as a product of prime factors.Completing the square is a method that is used to write a quadratic expression in a way such that it contains the perfect square.(i) The given expression is $x^2+12x+20$.Here, The coefficient of $x^2$ is $1$The coefficient of $x$ is $12$The constant term is $20$Coefficient of $x^2$ is $1$. So, we ... Read More

Given:The given quadratic polynomials are:(i) $4y^2+12y+5$(ii) $p^2+6p-16$To do:We have to factorize the given quadratic polynomials.Solution:Factorizing algebraic expressions:Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. An algebraic expression is factored completely when it is written as a product of prime factors.Completing the square is a method that is used to write a quadratic expression in a way such that it contains the perfect square.(i) The given expression is $4y^2+12y+5$.We can write $4y^2+12y+5$ as, $4y^2+12y+5=4(y^2+3y+\frac{5}{4})$Here, The coefficient of $y^2$ is $1$The coefficient of $y$ is $3$The constant term is $\frac{5}{4}$Coefficient of ... Read More

Given:The given quadratic polynomials are:(i) $p^2+6p+8$(ii) $q^2-10q+21$To do:We have to factorize the given quadratic polynomials.Solution:Factorizing algebraic expressions:Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. An algebraic expression is factored completely when it is written as a product of prime factors.Completing the square is a method that is used to write a quadratic expression in a way such that it contains the perfect square.(i) The given expression is $p^2+6p+8$.Here, The coefficient of $p^2$ is $1$The coefficient of $p$ is $6$The constant term is $8$Coefficient of $p^2$ is $1$. So, we ... Read More

Given:The given quadratic trinomials are:(i) $(x-2y)^2-5(x-2y)+6$(ii) $(2a-b)^2+2(2a-b)-8$To do:We have to factorize the given quadratic trinomials.Solution:Factorizing algebraic expressions:Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. An algebraic expression is factored completely when it is written as a product of prime factors.(i) The given expression is $(x-2y)^2-5(x-2y)+6$.We can factorize the given expression by splitting the middle term. Splitting the middle term means we have to rewrite the middle term as the sum or difference of the two terms.Here, The coefficient of $(x-2y)^2$ is $1$The coefficient of $(x-2y)$ is $-5$The constant ... Read More

Given:The given quadratic trinomials are:(i) $36a^2+12abc-15b^2c^2$(ii) $15x^2-16xyz-15y^2z^2$To do:We have to factorize the given quadratic trinomials.Solution:Factorizing algebraic expressions:Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. An algebraic expression is factored completely when it is written as a product of prime factors.(i) The given expression is $36a^2+12abc-15b^2c^2$.We can factorize the given expression by splitting the middle term. Splitting the middle term means we have to rewrite the middle term as the sum or difference of the two terms.Here, The coefficient of $a^2$ is $36$The coefficient of $a$ is $12bc$The constant ... Read More

Given:The given quadratic trinomials are:(i) $6x^2-13xy+2y^2$(ii) $14x^2+11xy-15y^2$(iii) $6a^2+17ab-3b^2$To do:We have to factorize the given quadratic trinomials.Solution:Factorizing algebraic expressions:Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. An algebraic expression is factored completely when it is written as a product of prime factors.(i) The given expression is $6x^2-13xy+2y^2$.We can factorize the given expression by splitting the middle term. Splitting the middle term means we have to rewrite the middle term as the sum or difference of the two terms.Here, The coefficient of $x^2$ is $6$The coefficient of $x$ is $-13y$The ... Read More