Questions: Simplify (2x^3-12x^2+16x)/(x^2+3x-28) ÷ (4x^2-36x+56)/(2x^2-98)

Solution Steps

We begin by factorizing the polynomials involved in the expression:

1. \( 2x^3 - 12x^2 + 16x = 2x(x - 4)(x - 2) \)
2. \( x^2 + 3x - 28 = (x - 4)(x + 7) \)
3. \( 4x^2 - 36x + 56 = 4(x - 7)(x - 2) \)
4. \( 2x^2 - 98 = 2(x - 7)(x + 7) \)

Next, we perform the division of the first polynomial by the second:

\[ \frac{2x^3 - 12x^2 + 16x}{x^2 + 3x - 28} \]

1. Divide \( 2x^3 \) by \( x^2 \), resulting in \( 2x \), with a remainder of \( -18x^2 + 72x \).
2. Divide \( -18x^2 \) by \( x^2 \), resulting in \( -18 \), with a remainder of \( 126x - 504 \).

Thus, the result of the first division is:

\[ \frac{2x^3 - 12x^2 + 16x}{x^2 + 3x - 28} = 2x + \frac{126x - 504}{x^2 + 3x - 28} - 18 \]

Now, we perform the division of the third polynomial by the fourth:

\[ \frac{2x^2 - 98}{4x^2 - 36x + 56} \]

1. Divide \( 2x^2 \) by \( 4x^2 \), resulting in \( \frac{1}{2} \), with a remainder of \( 18x - 126 \).

Thus, the result of the second division is:

\[ \frac{2x^2 - 98}{4x^2 - 36x + 56} = \frac{18x - 126}{4x^2 - 36x + 56} + \frac{1}{2} \]

The final expression can be constructed by combining the results of the two divisions. The complete expression is:

\[ \left(2x + \frac{126x - 504}{x^2 + 3x - 28} - 18\right) \cdot \left(\frac{18x - 126}{4x^2 - 36x + 56} + \frac{1}{2}\right) \]

This represents the simplified form of the original expression.

Final Answer