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Answer 1

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Answer 2

Step-by-step explanation:

Q.1

$12ax + 6 {b}^{2} - 4ab - 18bx$

Factor out 2 from the expression

$2(6ax + 3 {b}^{2} - 2ab - 9bx)$

Factor out 2a from the expression

$2(2a \times (3x - b) + 3 {b}^{2} - 9bx)$

Factor out -3b from the expression

$2(2a \times (3x - b) - 3b \times ( - b + 3x))$

Factor out 3x - b from the expression

Solution :

$2(3x - b) \times (2a - 3b)$

Q.2

$36 {a}^{2} {b}^{2} {c}^{2} - 54 {a}^{3} {b}^{3} {c}^{3} + 72 {a}^{4} {b}^{4} {c}^{4}$

Factor out 18a²b²c² from the expression

Solution :

$18 {a}^{2} {b}^{2} {c}^{2} \times (2 - 3abc + 4 {a}^{2} {b}^{2} {c}^{2} )$

Q.3

${x}^{2} - 11x + 30$

Write -11x as a difference

${x}^{2} - 5x - 6x + 30$

Factor out x from the expression

$x \times (x - 5) - 6x + 30$

Factor out -6 from the expression

$x \times (x - 5) - 6(x - 5)$

Factor out x - 5 from the expression

Solution :

$(x - 5) \times (x - 6)$

Q.4

$\frac{25 {x}^{2} {y}^{2} {z}^{2}( {x}^{2} - 14x - 51) }{5xyz(x - 17)( {x}^{2} - 9)}$

Write -14x as a difference

$\frac{25 {x}^{2} {y}^{2} {z}^{2} \times ( {x}^{2} + 3x - 17x - 51) }{5xyz \times (x - 17) \times ( {x}^{2} - 9) }$

Using a² - b² = ( a - b ) ( a + b ) , factor the expression

$\frac{25 {x}^{2} {y}^{2} {z}^{2} \times ( {x}^{2} + 3x - 17x - 51) }{5xyz \times (x - 17) \times (x - 3) \times (x + 3)}$

Reduce the fraction with 5

$\frac{5 {x}^{2} {y}^{2} {z}^{2} \times ( {x}^{2} + 3x - 17x - 51) }{xyz \times (x - 17) \times (x - 3) \times (x + 3)}$

Simplify the expression

$\frac{5xyz \times ( {x}^{2} + 3x - 17x - 51) }{(x - 17) \times (x - 3) \times (x + 3)}$

Factor out x from the expression

$\frac{5xyz \times (x(x + 3) - 17x - 51)}{(x - 17) \times (x - 3) \times (x + 3)}$

Factor out -17 from the expression

$\frac{5xyz \times (x \times (x + 3) - 17(x + 3))}{(x - 17) \times (x - 3) \times (x + 3)}$

Factor out x + 3 from the expression

$\frac{5xyz \times (x + 3) \times (x - 17)}{(x - 17) \times (x - 3) \times (x + 3)}$

Reduce the fraction with x + 3

$\frac{5xyz \times (x - 17)}{(x - 17) \times (x - 3)}$

Reduce the fraction with x - 17

Solution :

$\frac{5xyz}{x - 3}$

Q.5

$16 {a}^{4} - \frac{1}{81}$

Divide both sides of the equation by 16

${a}^{4} = \frac{1}{1296}$

Take the root of both sides of the equation and remember to use both positive and negative roots

$a = + and - \: \: \frac{1}{6}$

Write the solutions, one with + sign & one with

- sign

$a = - \frac{1}{6} \\ \\ a = \frac{1}{6}$

The equation has 2 solutions

$a_{1} = - \frac{1}{6} \: \: and \: \: a_{2} = \frac{1}{6}$

Q.6

${p}^{4} - {(p - q)}^{4}$

Using a² - b² = (a-b)(a+b) , factor the expression

$= ( {p}^{2} - (p - q) ^{2} \times ( {p}^{2} + (p - q) ^{2} )$

$= (p - (p - q)) \times (p + (p - q)) \times ( {p}^{2} + (p - q) ^{2} )$

Using (a-b)² = a²-2ab+b² , expand the expression

$(p - (p - q)) \times (p + (p - q)) \times ( {p}^{2} + {p}^{2} - 2pq + {q}^{2} )$

When there is - in front of an expression in parentheses, change the sign of each term in the expression

$(p - p + q) \times (p + (p - q)) \times ( {p}^{2} + {p}^{2} - 2pq + {q}^{2} )$

Collect like terms

$(p - p + q) \times (p + p - q) \times (2 {p}^{2} - 3pq + {q}^{2})$

Since two opposites add up to zero, remove them from the expression

$q \times (p + p - q) \times (2 {p}^{2} - 2pq + {q}^{2} )$

Collect like terms

Solution :

$q \times (2p - q) \times (2 {p}^{2} - 2pq + {q}^{2} )$

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