Algebraic Expressions and Identities — Mental Maths

Fill in the blanks:

(i) A symbol which has a fixed value is called a .....

(ii) A symbol which can be given various numerical values is called ..... or .....

(iii) The various parts of an algebraic expression separated by + or - sign are called .....

(iv) An algebraic expression having ..... terms is called a binomial.

(v) Each of the quantity (constant or literal) multiplied together to form a product is called a ..... of the product.

(vi) The terms having same literal coefficients are called ..... otherwise they are called .....

(vii) Degree of the polynomial is the greatest sum of the powers of ..... in each term.

(viii) An identity is an equality which is true for ..... of variables in it.

(ix) $(a + b)^2$ = .....

(x) $(x + a)(x + b) = x^2 + ..... + ab$

(xi) Dividend = ..... + remainder

Answer

(i) A symbol which has a fixed value is called a constant.

(ii) A symbol which can be given various numerical values is called variable or literal.

(iii) The various parts of an algebraic expression separated by + or - sign are called terms.

(iv) An algebraic expression having two terms is called a binomial.

(v) Each of the quantity (constant or literal) multiplied together to form a product is called a factor of the product.

(vi) The terms having same literal coefficients are called like terms otherwise they are called unlike terms.

(vii) Degree of the polynomial is the greatest sum of the powers of variables in each term.

(viii) An identity is an equality which is true for all values of variables in it.

(ix) $(a + b)^2 = \bm{a^2 + 2ab + b^2}$.

(x) $(x + a)(x + b) = x^2 + \bm{(a + b)x} + ab$.

(xi) Dividend = divisor × quotient + remainder.

State whether the following statements are true (T) or false (F):

(i) An algebraic expression having only one term is called a monomial.

(ii) A symbol which has fixed value is called a literal.

(iii) The term of an algebraic expression having no literal factor is called its constant term.

(iv) In any term of an algebraic expression the constant part is called literal coefficient of the term.

(v) An algebraic expression is called polynomial if the powers of the variables involved in it in each term are non-negative integers.

(vi) $5x^2y^2z, 3x^2zy^2, -\dfrac{4}{5}zx^2y^2$ are unlike terms.

(vii) $5x + \dfrac{2}{x} + 7$ is a polynomial of degree 1

(viii) $3x^2y + 7x + 8y + 9$ is a polynomial of degree 3

(ix) Numerical coefficient of $-7x^3y$ is 7

(x) $xyz + yzx + zxy$ is a monomial.

(xi) An equation is true for all values of variables in it.

(xii) $(a - b)^2 + 2ab = a^2 + b^2$

Answer

(i) An algebraic expression having only one term is called a monomial.

True.

(ii) A symbol which has fixed value is called a literal.

False.

Reason — A symbol which has a fixed value is called a constant. A literal is a symbol which can be given various numerical values.

(iii) The term of an algebraic expression having no literal factor is called its constant term.

True.

(iv) In any term of an algebraic expression the constant part is called literal coefficient of the term.

False.

Reason — In any term the constant part is called the numerical coefficient of the term, while the remaining part is called the literal coefficient.

(v) An algebraic expression is called polynomial if the powers of the variables involved in it in each term are non-negative integers.

True.

(vi) $5x^2y^2z, 3x^2zy^2, -\dfrac{4}{5}zx^2y^2$ are unlike terms.

False.

Reason — The literal coefficient of each of the three terms is $x^2y^2z$. As the terms have the same literal coefficients, they are like terms.

(vii) $5x + \dfrac{2}{x} + 7$ is a polynomial of degree 1

False.

Reason — Here $\dfrac{2}{x} = 2x^{-1}$, in which the power of $x$ is -1, a negative integer. So, the expression is not a polynomial at all.

(viii) $3x^2y + 7x + 8y + 9$ is a polynomial of degree 3

True.

(ix) Numerical coefficient of $-7x^3y$ is 7

False.

Reason — The numerical coefficient of $-7x^3y$ is -7.

(x) $xyz + yzx + zxy$ is a monomial.

True.

(xi) An equation is true for all values of variables in it.

False.

Reason — An equation is true only for certain values of the variables in it. An identity is true for all values of the variables in it.

(xii) $(a - b)^2 + 2ab = a^2 + b^2$

True.