Geometry. The volume of the box is represented by (x^2+5x+6)(x+5). Find the polynomial that represents the area of the bottom of the box.
the diagram has the height being x+2
My answer is: A=(x+5)(x+3)
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Multiply the following fractions
(2x^2-x-3)/(3x^2+7x+4) TIMES (3x^2-11x-20)/(4x^2-9)
My answer is: (x-5)/(2x+3)
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Divide the following fractions
(2m^2-5m-7)/(4m^2-9) divided by (5m^2+5m)/(2m^2+3m)
My answer is: (2m-7)/5(2m-3)
Looks OK!
To find the polynomial that represents the area of the bottom of the box, we need to divide the volume polynomial by the height polynomial.
The volume polynomial is (x^2+5x+6)(x+5) and the height is x+2.
To find the polynomial representing the area of the bottom, we divide the volume polynomial by the height polynomial:
Area = (x^2+5x+6)(x+5) / (x+2)
We can use long division or synthetic division to divide these polynomials. Let's use synthetic division.
Convert the volume polynomial into its standard form by arranging the terms in descending order:
(x^2+5x+6) = x^2 + 5x + 6 + 0x
Perform synthetic division:
-5 | 1 5 6
|_________
| -5 0
|_________
1 0 6
The remaining part after division is the polynomial representing the area of the bottom:
Area = x + 0 = x
Therefore, the polynomial that represents the area of the bottom of the box is A = x.
For the multiplication of fractions:
(2x^2-x-3)/(3x^2+7x+4) * (3x^2-11x-20) / (4x^2-9)
To multiply fractions, we multiply the numerators and denominators separately:
Numerator = (2x^2-x-3) * (3x^2-11x-20)
Denominator = (3x^2+7x+4) * (4x^2-9)
Multiply the numerators and denominators:
Numerator = (2x^2-x-3) * (3x^2-11x-20)
Denominator = (3x^2+7x+4) * (4x^2-9)
Simplify each numerator and denominator by factoring if possible:
Numerator = (2x + 1)(x - 3)(3x + 4)(3x - 5)
Denominator = (3x + 4)(x + 1)(2x + 3)(2x - 3)
Cancel out common factors between numerator and denominator:
Numerator = (2x + 1)(x - 3)(3x - 5)
Denominator = (x + 1)(2x + 3)(2x - 3)
Therefore, the result is:
(2x^2-x-3)/(3x^2+7x+4) * (3x^2-11x-20) / (4x^2-9) = (2x + 1)(x - 3)(3x - 5) / (x + 1)(2x + 3)(2x - 3)
For the division of fractions:
(2m^2-5m-7)/(4m^2-9) ÷ (5m^2+5m)/(2m^2+3m)
To divide fractions, we multiply the first fraction by the reciprocal (inverse) of the second fraction:
(2m^2-5m-7)/(4m^2-9) * (2m^2+3m)/(5m^2+5m)
To simplify this, factor the numerators and denominators if possible:
Numerator = (2m^2-5m-7) * (2m^2+3m)
Denominator = (4m^2-9) * (5m^2+5m)
Factor the quadratics:
Numerator = (m - 7)(2m + 1)(m + 7)(2m + 3)
Denominator = (2m - 3)(2m + 3)(5m^2+5m)
Cancel out common factors between numerator and denominator:
Numerator = (m - 7)(2m + 1)(m + 7)
Denominator = (2m - 3)(5m^2+5m)
Therefore, the result is:
(2m^2-5m-7)/(4m^2-9) ÷ (5m^2+5m)/(2m^2+3m) = (m - 7)(2m + 1)(m + 7) / (2m - 3)(5m^2+5m)