DAV Class 7 Maths Chapter 6 Worksheet 5
Algebraic Expression Worksheet 5
1. Express the following as a product of its any two factors (in four different ways).
(i) \( 12x^2y \)
\begin{align*} 4x^2y & \times 3 \\ 2x^2 & \times 6y \\ 3xy & \times 4x \\ 2x & \times 6xy \end{align*}
(ii) \( 18ab^2 \)
\begin{align*} 9a & \times 2b^2 \\ 18a & \times b^2 \\ 3ab & \times 6b \\ 6a & \times 3b^2 \\ \end{align*}
(iii) \( 24c^2b \)
\begin{align*} 2c & \times 12cb \\ 3cb & \times 8c \\ 4b & \times 6c^2 \\ 24c & \times cb \\ \end{align*}
2. Find the H.C.F of the following monomials
(i) \( 2a^5 \text{ and } 12 a^2 \)
\begin{align*} 2a^5 &= \boxed{2} \times \boxed{a} \times \boxed{a} \times a \times a \times a \\ 12a^2 &= \boxed{2} \times 2 \times 3 \times \boxed{a} \times \boxed{a} \\ \text{Common factors} & = 2 , a , a \\ \text{HCF }& = 2 \times a \times a \implies 2a^2 \end{align*}
Answer \( \color{red} 2a^2\)
(ii) \( 9x^3y \text{ and } 18x^2y^3 \)
\[ \begin{align*} 9x^3y &= \boxed{3} \times \boxed{3} \times \boxed{x} \times \boxed{x} \times x \times \boxed{y} \\ 18x^2y^3 &= \boxed{3} \times \boxed{3} \times 2 \times \boxed{x} \times \boxed{x} \times \boxed{y} \times y \times y \\ \text{Common factors} & = 3, 3, x, x, y \\ \text{HCF }& = 9 \times x \times x \times y \implies 9x^2y \end{align*} \]
Answer \( \color{red} 9x^2y \)
(iii) \( a^2b^3 \text{ and } a^3b^2 \)
\[ \begin{align*} a^2b^3 &= \boxed{a} \times \boxed{a} \times \boxed{b} \times \boxed{b} \times b \\ a^3b^2 &= \boxed{a} \times \boxed{a} \times a \times \boxed{b} \times \boxed{b} \\ \text{Common factors} & = a, a, b, b \\ \text{HCF }& = a \times a \times b \times b \implies a^2b^2 \end{align*} \]
Answer \( \color{red} a^2b^2 \)
(iv) \( 15a^3, -45a^2, 150a \)
\[ \begin{align*} 15a^3 &= \boxed{3} \times \boxed{5} \times a \times \boxed{a} \times a \\ -45a^2 &= -3 \times \boxed{3} \times \boxed{5} \times \boxed{a} \times a \\ 150a &= \boxed{3} \times \boxed{5} \times 5 \times 2 \times \boxed{a} \\ \text{Common factors} & = 3, 5, a \\ \text{HCF }& = 3 \times 5 \times a \implies 15a \end{align*} \]
Answer \( \color{red} 15a \)
(v) \( 2x^3y^2, 10x^2y^3, 14xy \)
\[ \begin{align*} 2x^3y^2 &= \boxed{2} \times \boxed{x} \times x \times x \times \boxed{y} \times y \\ 10x^2y^3 &= \boxed{2} \times 5 \times \boxed{x} \times x \times \boxed{y} \times y \times y \\ 14xy &= \boxed{2} \times 7 \times \boxed{x} \times \boxed{y} \\ \text{Common factors} & = 2, x, y \\ \text{HCF }& = 2 \times x \times y \implies 2xy \end{align*} \]
Answer \( \color{red} 2xy \)
(vi) \( x^3y^2 , -8y^2 \)
\[ \begin{align*} x^3y^2 &= x \times x \times x \times \boxed{y} \times \boxed{y} \\ -8y^2 &= -2 \times 2 \times 2 \times \boxed{y} \times \boxed{y} \\ \text{Common factors} & = y, y \\ \text{HCF }& = y \times y \implies y^2 \end{align*} \]
Answer \( \color{red} y^2 \)
3. Find the H.C.F of the terms and factorise
(i) \( 5y - 15y^2 \)
\begin{align*} 5y &= \boxed{5} \times \boxed{y} \\ - 15y^2 &= -3 \times \boxed{5} \times \boxed{y} \times y \\ \text{HCF }& = 5y \\ \\ 5y - 15y^2 &= ({\color{magenta}5y} \times 1) - ({\color{magenta}5y} \times 3y) \\ & = {\color{magenta}5y}(1 - 3y) \end{align*}
Answer \( \color{red} 5y(1 - 3y) \)
(ii) \( 16m - 4m^2 \)
\[ \begin{align*} 16m &= \boxed{4} \times \boxed{m} \times 4 \\ -4m^2 &= -\boxed{4} \times \boxed{m} \times m \\ \text{HCF }& = 4m \\ \\ 16m - 4m^2 &= ({\color{magenta}4m} \times 4) - ({\color{magenta}4m} \times m) \\ & = {\color{magenta}4m}(4 - m) \end{align*} \]
Answer \( \color{red} 4m(4 - m) \)
(iii) \( 8x^3y^2 + 8x^3 \)
\[ \begin{align*} 8x^3y^2 &= \boxed{8} \times \boxed{x^3} \times y^2 \\ 8x^3 &= \boxed{8} \times \boxed{x^3} \\ \text{HCF }& = 8x^3 \\ \\ 8x^3y^2 + 8x^3 &= ({\color{magenta}8x^3} \times y^2) + ({\color{magenta}8x^3} \times 1) \\ & = {\color{magenta}8x^3}(y^2 + 1) \end{align*} \]
Answer \( \color{red} 8x^3(y^2 + 1) \)
(iv) \( 20x^3 - 40x^2 + 80x \)
\[ \begin{align*} 20x^3 &= \boxed{20} \times \boxed{x} \times x \times x \\ -40x^2 &= -\boxed{20} \times 2 \times \boxed{x} \times x \\ 80x &= \boxed{20} \times 4 \times \boxed{x} \\ \text{HCF }& = 20x \\ \\ 20x^3 - 40x^2 + 80x &= ({\color{magenta}20x} \times x^2) - ({\color{magenta}20x} \times 2x) + ({\color{magenta}20x} \times 4) \\ & = {\color{magenta}20x}(x^2 - 2x + 4) \end{align*} \]
Answer \( \color{red} 20x(x^2 - 2x + 4) \)
(v) \( x^4y - 3x^2y^2 - 6xy^3 \)
\[ \begin{align*} x^4y &= \boxed{x} \times x \times x \times x \times \boxed{y} \\ -3x^2y^2 &= -3 \times \boxed{x} \times x \times \boxed{y} \times y \\ -6xy^3 &= -2 \times 3 \times \boxed{x} \times \boxed{y} \times y \times y \\ \text{HCF }& = x \times y \\ \\ x^4y - 3x^2y^2 - 6xy^3 &= ({\color{magenta}xy} \times x^3) - ({\color{magenta}xy} \times 3xy) - ({\color{magenta}xy} \times 6y^2) \\ & = {\color{magenta}xy}(x^3 - 3xy - 6y^2) \end{align*} \]
Answer \( \color{red} xy(x^3 - 3xy - 6y^2) \)
(vi) \( 8x^2y^2 - 16xy^3 + 24xy \)
\[ \begin{align*} 8x^2y^2 &= \boxed{8} \times \boxed{x} \times x \times \boxed{y} \times y \\ -16xy^3 &= -2 \times \boxed{8} \times \boxed{x} \times \boxed{y} \times y \times y \\ 24xy &= 3 \times \boxed{8} \times \boxed{x} \times \boxed{y} \\ \text{HCF }& = 8xy \\ \\ 8x^2y^2 - 16xy^3 + 24xy &= ({\color{magenta}8xy} \times xy) - ({\color{magenta}8xy} \times 2y^2) + ({\color{magenta}8xy} \times 3) \\ & = {\color{magenta}8xy}(xy - 2y^2 + 3) \end{align*} \]
Answer \( \color{red} 8xy(xy - 2y^2 + 3) \)