Answer :
To solve the cubic equation [tex]\(48x^3 + 40x^2 - x - 3 = 0\)[/tex] given that [tex]\(-\frac{3}{4}\)[/tex] is a zero, we can use polynomial division to factor the cubic polynomial and find the remaining roots.
### Step-by-step solution:
1. Given Zero: We know that [tex]\(-\frac{3}{4}\)[/tex] is a root of the polynomial. This means that [tex]\(x = -\frac{3}{4}\)[/tex] is a solution, and [tex]\((4x + 3)\)[/tex] is a factor of the polynomial.
2. Polynomial Division:
- We can perform synthetic or long division to divide the polynomial [tex]\(48x^3 + 40x^2 - x - 3\)[/tex] by [tex]\((4x + 3)\)[/tex].
- After performing the division, the quotient will be a quadratic polynomial.
3. Find the Quotient:
- After dividing, the quotient comes out to be a quadratic polynomial: [tex]\(12x^2 + 10x - 1\)[/tex].
4. Solve the Quadratic Equation: Now we focus on [tex]\(12x^2 + 10x - 1 = 0\)[/tex].
- Use the quadratic formula to find the roots:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 12\)[/tex], [tex]\(b = 10\)[/tex], and [tex]\(c = -1\)[/tex].
5. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 10^2 - 4 \times 12 \times (-1) = 100 + 48 = 148
\][/tex]
6. Find the Roots:
- [tex]\[
x = \frac{-10 \pm \sqrt{148}}{24}
\][/tex]
- Simplify [tex]\(\sqrt{148}\)[/tex] which is [tex]\(2\sqrt{37}\)[/tex], so:
[tex]\[
x = \frac{-10 \pm 2\sqrt{37}}{24}
\][/tex]
- Simplifying further by dividing numerator and denominator by 2:
[tex]\[
x = \frac{-5 \pm \sqrt{37}}{12}
\][/tex]
7. Real Roots: Evaluate these to get the approximate decimal values:
- The values lead to: [tex]\(-\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex] for the given set of solutions.
8. Solution Set:
- Therefore, the solution set of the equation [tex]\(48x^3 + 40x^2 - x - 3 = 0\)[/tex] is:
[tex]\[
-\frac{3}{4}, -\frac{1}{3}, \frac{1}{4}
\][/tex]
These roots satisfy the original equation using the given information that [tex]\(-\frac{3}{4}\)[/tex] is already a known zero. The solution set is [tex]\(-\frac{3}{4}, -\frac{1}{3}, \frac{1}{4}\)[/tex].
### Step-by-step solution:
1. Given Zero: We know that [tex]\(-\frac{3}{4}\)[/tex] is a root of the polynomial. This means that [tex]\(x = -\frac{3}{4}\)[/tex] is a solution, and [tex]\((4x + 3)\)[/tex] is a factor of the polynomial.
2. Polynomial Division:
- We can perform synthetic or long division to divide the polynomial [tex]\(48x^3 + 40x^2 - x - 3\)[/tex] by [tex]\((4x + 3)\)[/tex].
- After performing the division, the quotient will be a quadratic polynomial.
3. Find the Quotient:
- After dividing, the quotient comes out to be a quadratic polynomial: [tex]\(12x^2 + 10x - 1\)[/tex].
4. Solve the Quadratic Equation: Now we focus on [tex]\(12x^2 + 10x - 1 = 0\)[/tex].
- Use the quadratic formula to find the roots:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 12\)[/tex], [tex]\(b = 10\)[/tex], and [tex]\(c = -1\)[/tex].
5. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 10^2 - 4 \times 12 \times (-1) = 100 + 48 = 148
\][/tex]
6. Find the Roots:
- [tex]\[
x = \frac{-10 \pm \sqrt{148}}{24}
\][/tex]
- Simplify [tex]\(\sqrt{148}\)[/tex] which is [tex]\(2\sqrt{37}\)[/tex], so:
[tex]\[
x = \frac{-10 \pm 2\sqrt{37}}{24}
\][/tex]
- Simplifying further by dividing numerator and denominator by 2:
[tex]\[
x = \frac{-5 \pm \sqrt{37}}{12}
\][/tex]
7. Real Roots: Evaluate these to get the approximate decimal values:
- The values lead to: [tex]\(-\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex] for the given set of solutions.
8. Solution Set:
- Therefore, the solution set of the equation [tex]\(48x^3 + 40x^2 - x - 3 = 0\)[/tex] is:
[tex]\[
-\frac{3}{4}, -\frac{1}{3}, \frac{1}{4}
\][/tex]
These roots satisfy the original equation using the given information that [tex]\(-\frac{3}{4}\)[/tex] is already a known zero. The solution set is [tex]\(-\frac{3}{4}, -\frac{1}{3}, \frac{1}{4}\)[/tex].
