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Problem 625

# Remove fractional coefficients from the equation $2 \mathrm{x}^{3}-(2 / 3) \mathrm{x}^{2}-(1 / 8) \mathrm{x}+(3 / 16)=0$

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To remove the fractional coefficients from the equation $$2x^3 - (\frac{2}{3})x^2 - (\frac{1}{8})x + (\frac{3}{16}) = 0$$, find the least common denominator (LCD) of the fractions, which is 24. Then, multiply the entire equation by the LCD and distribute it to each term: $$48x^3 - 16x^2 - 3x + 4.5 = 0$$. The equivalent equation without fractions is $$48x^3 - 16x^2 - 3x + 4.5 = 0$$.
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## Step 1: Identify the fractions and their denominators

First, we need to identify the fractional coefficients in the given equation and their respective denominators: $$2x^3 - (\frac{2}{3})x^2 - (\frac{1}{8})x + (\frac{3}{16}) = 0$$ The fractions are $$\frac{2}{3}$$, $$\frac{1}{8}$$, and $$\frac{3}{16}$$, and their denominators are 3, 8, and 16.

## Step 2: Find the least common denominator (LCD)

Now, we need to find the least common denominator (LCD) of the three identified denominators, which is the smallest positive integer divisible by the three denominators. The prime factors of the denominators are: 3: $$\underline{3}$$ 8: 2 × 2 × $$\underline{2}$$ 16: 2 × 2 × 2 × $$\underline{2}$$ Taking the highest power of each prime factor present in the prime factorizations, the LCD is: LCD = 2³ × 3 = 24

## Step 3: Multiply the equation by the LCD

Now, we will multiply the entire equation by the LCD, which is 24: 24(2x³ - ($$\frac{2}{3}$$x² - $$\frac{1}{8}$$x + $$\frac{3}{16}$$) = 24 × 0

## Step 4: Distribute the LCD

Next, distribute the LCD to each term in the equation: 48x³ - $$\frac{2}{3}$$ × 24x² - $$\frac{1}{8}$$ × 24x + $$\frac{3}{16}$$ × 24 = 0 Simplify the fractions and multiply: 48x³ - 16x² - 3x + 4.5 = 0

## Step 5: Rewrite the equation without fractions

Now, we have an equation without fractions as desired: 48x³ - 16x² - 3x + 4.5 = 0 The given equation with fractional coefficients has been transformed into an equivalent equation with no fractions.

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