Mathematics: Tickets to a fair cost 12 for each adult an 7 for each child write an expression to find the cost for 3 adults and 6 children

Tickets to a fair cost 12 for each adult an 7 for

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24.01.2023

1) The factorization is: 2a(a^2 + 2a + 4)

2) The factorization is: 6y^2(y - 7)(y + 7)

3) The factorization is: 5(x+5)^2

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To factor each expression, we apply the given concepts, that is, greatest common factor, difference of squares and perfect squares.

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Expression 1:

The expression is:

2a^3 + 4a^2 + 8a

First, we find the gcf of the numbers 2, 4 and 8, which is 2.

Of the exponents, the gcf between 3, 2 and 1 is 1, so:

$2a^3 + 4a^2 + 8a = 2a(\frac{2a^3}{2a} + \frac{4a^2}{2a} + \frac{8a}{2a}) = 2a(a^2 + 2a + 4)$

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Expression 2:

The expression is:

6y^4 - 294y^2

The gcf of 6 and 294 is 6.

The gcf of the exponents 2 and 4 is 4.

Thus:

$6y^4 - 294y^2 = 6y^2(\frac{6y^4}{6y^2} - \frac{294y^2}{6y^2}) = 6y^2(y^2 - 49)$

Then, applying the difference of squares:

y^2 - 49 = (y - 7)(y + 7)

Thus, the factored expression is:

6y^2(y^2 - 49) = 6y^2(y - 7)(y + 7)

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Expression 3:

The expression is:

5x^2 + 50x + 125

The gcf of the coefficients, 5, 50 and 125 is 5, so:

$5(\frac{5x^2}{5} + \frac{50x}{5} + \frac{125}{5}) = 5(x^2 + 10x + 25)$

Applying the perfect square, we get that:

x^2 + 10x + 25 = (x + 5)^2

Tus, the factored expression is:

5(x^2 + 10x + 25) = 5(x+5)^2

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