DAV Class 8 Maths Chapter 7 Worksheet 7
Algebraic Identities Worksheet 7
Factorize the following expressions:
1. \( x^2 + 14x + 33 \)
Solution
\[ \begin{align*} &= x^2 \ {\color{magenta} + \ 14x} + 33 \\[4pt] &= x^2 \ {\color{magenta} + \ 3x + 11x} + 33 \\[4pt] &= x(x + 3) + 11(x + 3) \\[4pt] & = \color{red} (x + 3)(x + 11) \end{align*} \]
2. \( y^2 - y - 6 \)
Solution
\[ \begin{align*} &= y^2 \ {\color{magenta} - \ y} - 6 \\[4pt] &= y^2 \ {\color{magenta} + \ 2y-3y } - 6 \\[4pt] &= y(y +2) -3(y +2) \\[4pt] & = \color{red} (y + 2)(y - 3) \end{align*} \]
3. \( x^2 + x - 72 \)
Solution
\[ \begin{align*} &= x^2 \ {\color{magenta} + \ x} - 72 \\[4pt] &= x^2 \ {\color{magenta} - \ 8x + 9x} - 72 \\[4pt] &= x(x - 8) + 9(x - 8) \\[4pt] & = \color{red} (x - 8)(x + 9) \end{align*} \]
4. \( x^2 + 12x + 27 \)
Solution
\[ \begin{align*} &= x^2 \ {\color{magenta} + \ 12x} + 27 \\[4pt] &= x^2 \ {\color{magenta} + \ 3x + 9x} + 27 \\[4pt] &= x(x + 3) + 9(x + 3) \\[4pt] & = \color{red} (x + 3)(x + 9) \end{align*} \]
5. \( y^2 + y - 132 \)
Solution
\[ \begin{align*} &= y^2 \ {\color{magenta} + \ y} - 132 \\[4pt] &= y^2 \ {\color{magenta} - \ 11y + 12 y} - 132 \\[4pt] &= y(y - 11) + 12(y - 11) \\[4pt] & = \color{red} (y - 11)(y + 12) \end{align*} \]
6. \( x^2 + 11x + 30 \)
Solution
\[ \begin{align*} &= x^2 \ {\color{magenta} + \ 11x} + 30 \\[4pt] &= x^2 \ {\color{magenta} + \ 5x + 6x} + 30 \\[4pt] &= x(x + 5) + 6(x + 5) \\[4pt] & = \color{red} (x + 5)(x + 6) \end{align*} \]
7. \( x^2 - 11x - 42 \)
Solution
\[ \begin{align*} &= x^2 \ {\color{magenta} - \ 11x} - 42 \\[4pt] &= x^2 \ {\color{magenta} + \ 3x-14x } - 42 \\[4pt] &= x(x +3) -14(x +3 ) \\[4pt] & = \color{red} (x + 3)(x - 14) \end{align*} \]
8. \( z^2 - 12z + 27 \)
Solution
\[ \begin{align*} &= z^2 \ {\color{magenta} - \ 12z} + 27 \\[4pt] &= z^2 \ {\color{magenta} - \ 3z - 9z} + 27 \\[4pt] &= z(z - 3) - 9(z - 3) \\[4pt] & = \color{red} (z - 3)(z - 9) \end{align*} \]
9. \( p^2 - 5p - 6 \)
Solution
\[ \begin{align*} &= p^2 \ {\color{magenta} - \ 5p} - 6 \\[4pt] &= p^2 \ {\color{magenta} - \ 6p + 1p} - 6 \\[4pt] &= p(p - 6) + 1(p - 6) \\[4pt] & = \color{red} (p + 1)(p - 6) \end{align*} \]
10. \( p^2 + p - 56 \)
Solution
\[ \begin{align*} &= p^2 \ {\color{magenta} + \ p} - 56 \\[4pt] &= p^2 \ {\color{magenta} + \ 8p - 7p} - 56 \\[4pt] &= p(p + 8) - 7(p + 8) \\[4pt] & = \color{red} (p - 7)(p + 8) \end{align*} \]
11. \( x^2 - 8x - 65 \)
Solution
\[ \begin{align*} &= x^2 \ {\color{magenta} - \ 8x} - 65 \\[4pt] &= x^2 \ {\color{magenta} - \ 13x + 5x} - 65 \\[4pt] &= x(x - 13) + 5(x - 13) \\[4pt] & = \color{red} (x + 5)(x - 13) \end{align*} \]
12. \( x^2 + 8x + 15 \)
Solution
\[ \begin{align*} &= x^2 \ {\color{magenta} + \ 8x} + 15 \\[4pt] &= x^2 \ {\color{magenta} + \ 5x + 3x} + 15 \\[4pt] &= x(x + 5) + 3(x + 5) \\[4pt] & = \color{red} (x + 3)(x + 5) \end{align*} \]
13. \( x^2 + 2x - 24 \)
Solution
\[ \begin{align*} &= x^2 \ {\color{magenta} + \ 2x} - 24 \\[4pt] &= x^2 \ {\color{magenta} + \ 6x - 4x} - 24 \\[4pt] &= x(x + 6) - 4(x + 6) \\[4pt] & = \color{red} (x - 4)(x + 6) \end{align*} \]
14. \( x^2 - 3x - 54 \)
Solution
\[ \begin{align*} &= x^2 \ {\color{magenta} - \ 3x} - 54 \\[4pt] &= x^2 \ {\color{magenta} - \ 9x + 6x} - 54 \\[4pt] &= x(x - 9) + 6(x - 9) \\[4pt] & = \color{red} (x + 6)(x - 9) \end{align*} \]
15. \( a^2 - 7a + 12 \)
Solution
\[ \begin{align*} &= a^2 \ {\color{magenta} - \ 7a} + 12 \\[4pt] &= a^2 \ {\color{magenta} - \ 4a - 3a} + 12 \\[4pt] &= a(a - 4) - 3(a - 4) \\[4pt] & = \color{red} (a - 3)(a - 4) \end{align*} \]
16. \( p^2 - 3pq + 2q^2 \)
Solution
\[ \begin{align*} &= p^2 \ {\color{magenta} - \ 3pq} + 2q^2 \\[4pt] &= p^2 \ {\color{magenta} - \ pq - 2pq} + 2q^2 \\[4pt] &= p(p - q) - 2q(p - q) \\[4pt] & = \color{red} (p - q)(p - 2q) \end{align*} \]