Factorisation — Mental Maths

Fill in the blanks:

(i) When an algebraic expression can be written as the product of two or more expressions then each of these expressions is called ............... of the given expression.

(ii) The process of finding two or more expressions whose product is the given expression is called ...............

(iii) HCF of two or more monomials = (HCF of their ............... coefficients) × (HCF of their literal coefficients)

(iv) HCF of literal coefficients = product of each common literal raised to the ............... power.

(v) To factorise the trinomial of the form $x^2 + px + q$, we need to find two integers $a$ and $b$ such that $a + b$ = ............... and $ab$ = ...............

(vi) To factorise the trinomial of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are integers, we split $b$ into two parts such that ............... of these parts is $b$ and their ............... is $ac$

Answer

(i) When an algebraic expression can be written as the product of two or more expressions then each of these expressions is called factor of the given expression.

(ii) The process of finding two or more expressions whose product is the given expression is called factorisation.

(iii) HCF of two or more monomials = (HCF of their numerical coefficients) × (HCF of their literal coefficients)

(iv) HCF of literal coefficients = product of each common literal raised to the lowest power.

(v) To factorise the trinomial of the form $x^2 + px + q$, we need to find two integers $a$ and $b$ such that $a + b$ = $\bm{p}$ and $ab$ = $\bm{q}$.

(vi) To factorise the trinomial of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are integers, we split $b$ into two parts such that algebraic sum of these parts is $b$ and their product is $ac$

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

(i) Factorisation is the reverse process of multiplication.

(ii) HCF of two or more polynomials (with integral coefficients) is the smallest common factor of the given polynomials.

(iii) HCF of $6x^2y^2$ and $8xy^3$ is $2xy^2$

(iv) Factorisation by grouping is possible only if the given polynomial contains an even number of terms.

(v) To factorise the trinomial of the form $ax^2 + bx + c$ where, $a$, $b$, $c$ are integers we want to find two integers $A$ and $B$ such that $A + B = ac$ and $AB = b$

(vi) Factors of $4x^2 - 12x + 9$ are $(2x - 3)(2x - 3)$

(vii) The value of $p$ for $(51)^2 - (49)^2 = 100p$ is 2

(viii) The difference of the squares of two consecutive natural numbers is their sum.

Answer

(i) Factorisation is the reverse process of multiplication.

True.

(ii) HCF of two or more polynomials (with integral coefficients) is the smallest common factor of the given polynomials.

False.

Reason — HCF of two or more polynomials (with integral coefficients) is the largest common factor of the given polynomials.

(iii) HCF of $6x^2y^2$ and $8xy^3$ is $2xy^2$

True.

(iv) Factorisation by grouping is possible only if the given polynomial contains an even number of terms.

True.

(v) To factorise the trinomial of the form $ax^2 + bx + c$ where, $a$, $b$, $c$ are integers we want to find two integers $A$ and $B$ such that $A + B = ac$ and $AB = b$

False.

Reason — We want to find two integers $A$ and $B$ such that $A + B = b$ and $AB = ac$.

(vi) Factors of $4x^2 - 12x + 9$ are $(2x - 3)(2x - 3)$

True.

(vii) The value of $p$ for $(51)^2 - (49)^2 = 100p$ is 2

True.

(viii) The difference of the squares of two consecutive natural numbers is their sum.

True.