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Answer :
To solve the equation [tex]\(48x^3 + 40x^2 - x - 3 = 0\)[/tex] knowing that [tex]\(-\frac{3}{4}\)[/tex] is a zero, we can use polynomial division to simplify the expression and find the other roots.
### Step 1: Use Polynomial Division
Since [tex]\(-\frac{3}{4}\)[/tex] is a root, we can factor the polynomial as [tex]\((x + \frac{3}{4})(\text{some quadratic polynomial}) = 0\)[/tex].
Convert the root for easier division:
Since [tex]\(-\frac{3}{4}\)[/tex] is a root, this means [tex]\(x + \frac{3}{4}\)[/tex] divides the polynomial. However, for synthetic division, it's more practical to work with integer coefficients. Hence, multiply the entire expression by 4 to get integer coefficients for simplification, turning the root to [tex]\(-3\)[/tex] and the divisor to [tex]\(4x + 3\)[/tex].
Perform Synthetic Division:
- Divide [tex]\(48x^3 + 40x^2 - x - 3\)[/tex] by [tex]\(4x + 3\)[/tex].
- The division will yield a quadratic polynomial [tex]\(48x^2 + 6x - 4\)[/tex].
### Step 2: Solve the Quadratic Equation
The polynomial now becomes:
[tex]\((4x + 3)(48x^2 + 6x - 4) = 0\)[/tex].
We already know one root is [tex]\(-\frac{3}{4}\)[/tex] from solving [tex]\(4x + 3 = 0\)[/tex]. For the quadratic [tex]\(48x^2 + 6x - 4 = 0\)[/tex], use the quadratic formula:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\][/tex]
where [tex]\(a = 48\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = -4\)[/tex].
Calculate the Discriminant:
[tex]\[b^2 - 4ac = 6^2 - 4 \cdot 48 \cdot (-4) = 36 + 768 = 804.\][/tex]
Find the roots of the quadratic equation:
1. [tex]\(x_1 = \frac{-6 + \sqrt{804}}{2 \cdot 48}\)[/tex]
2. [tex]\(x_2 = \frac{-6 - \sqrt{804}}{2 \cdot 48}\)[/tex]
This simplifies approximately to:
1. [tex]\(x_1 = 0.25\)[/tex]
2. [tex]\(x_2 = -0.333...\)[/tex] (or [tex]\(-\frac{1}{3}\)[/tex])
### Solution Set
Thus, the solutions to the equation [tex]\(48x^3 + 40x^2 - x - 3 = 0\)[/tex] are:
[tex]\(-\frac{3}{4}, 0.25, -\frac{1}{3}\)[/tex].
The solution set is [tex]\(-\frac{3}{4}, 0.25, -0.33333\ldots\)[/tex]
### Step 1: Use Polynomial Division
Since [tex]\(-\frac{3}{4}\)[/tex] is a root, we can factor the polynomial as [tex]\((x + \frac{3}{4})(\text{some quadratic polynomial}) = 0\)[/tex].
Convert the root for easier division:
Since [tex]\(-\frac{3}{4}\)[/tex] is a root, this means [tex]\(x + \frac{3}{4}\)[/tex] divides the polynomial. However, for synthetic division, it's more practical to work with integer coefficients. Hence, multiply the entire expression by 4 to get integer coefficients for simplification, turning the root to [tex]\(-3\)[/tex] and the divisor to [tex]\(4x + 3\)[/tex].
Perform Synthetic Division:
- Divide [tex]\(48x^3 + 40x^2 - x - 3\)[/tex] by [tex]\(4x + 3\)[/tex].
- The division will yield a quadratic polynomial [tex]\(48x^2 + 6x - 4\)[/tex].
### Step 2: Solve the Quadratic Equation
The polynomial now becomes:
[tex]\((4x + 3)(48x^2 + 6x - 4) = 0\)[/tex].
We already know one root is [tex]\(-\frac{3}{4}\)[/tex] from solving [tex]\(4x + 3 = 0\)[/tex]. For the quadratic [tex]\(48x^2 + 6x - 4 = 0\)[/tex], use the quadratic formula:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\][/tex]
where [tex]\(a = 48\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = -4\)[/tex].
Calculate the Discriminant:
[tex]\[b^2 - 4ac = 6^2 - 4 \cdot 48 \cdot (-4) = 36 + 768 = 804.\][/tex]
Find the roots of the quadratic equation:
1. [tex]\(x_1 = \frac{-6 + \sqrt{804}}{2 \cdot 48}\)[/tex]
2. [tex]\(x_2 = \frac{-6 - \sqrt{804}}{2 \cdot 48}\)[/tex]
This simplifies approximately to:
1. [tex]\(x_1 = 0.25\)[/tex]
2. [tex]\(x_2 = -0.333...\)[/tex] (or [tex]\(-\frac{1}{3}\)[/tex])
### Solution Set
Thus, the solutions to the equation [tex]\(48x^3 + 40x^2 - x - 3 = 0\)[/tex] are:
[tex]\(-\frac{3}{4}, 0.25, -\frac{1}{3}\)[/tex].
The solution set is [tex]\(-\frac{3}{4}, 0.25, -0.33333\ldots\)[/tex]
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