To solve the equation [tex]\(x^3 - 5x^2 + 2 = -x^3 + 17x\)[/tex], let's start by simplifying and combining like terms.
Step 1: Start with the given equation:
[tex]\[ x^3 - 5x^2 + 2 = -x^3 + 17x \][/tex]
Step 2: Move all terms to one side to set the equation to 0. Add [tex]\(x^3\)[/tex] and subtract [tex]\(17x\)[/tex] from both sides:
[tex]\[ x^3 - 5x^2 + 2 + x^3 - 17x = 0 \][/tex]
[tex]\[ 2x^3 - 5x^2 - 17x + 2 = 0 \][/tex]
Now we have a polynomial equation:
[tex]\[ 2x^3 - 5x^2 - 17x + 2 = 0 \][/tex]
Step 3: Solve the polynomial equation. This involves finding the roots of the polynomial [tex]\(2x^3 - 5x^2 - 17x + 2\)[/tex].
Finding the exact roots of a cubic equation typically requires numerical methods or algebraic manipulation. We can use techniques such as the Rational Root Theorem to test possible rational roots, and then use polynomial division.
Step 4: Test possible rational roots. The Rational Root Theorem states that any rational root of the polynomial [tex]\(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\)[/tex] is a factor of the constant term [tex]\(a_0\)[/tex] divided by a factor of the leading coefficient [tex]\(a_n\)[/tex].
For our polynomial [tex]\(2x^3 - 5x^2 - 17x + 2\)[/tex]:
- The constant term [tex]\(a_0 = 2\)[/tex]
- The leading coefficient [tex]\(a_n = 2\)[/tex]
The possible rational roots are therefore:
[tex]\[ \pm 1, \pm 2 \][/tex]
Step 5: Test these possible rational roots by substituting them into the polynomial:
- For [tex]\(x = 1\)[/tex]:
[tex]\[ 2(1)^3 - 5(1)^2 - 17(1) + 2 = 2 - 5 - 17 + 2 = -18 \neq 0 \][/tex]
- For [tex]\(x = -1\)[/tex]:
[tex]\[ 2(-1)^3 - 5(-1)^2 - 17(-1) + 2 = -2 - 5 + 17 + 2 = 12 \neq 0 \][/tex]
- For [tex]\(x = 2\)[/tex]:
[tex]\[ 2(2)^3 - 5(2)^2 - 17(2) + 2 = 16 - 20 - 34 + 2 = -36 \neq 0 \][/tex]
- For [tex]\(x = -2\)[/tex]:
[tex]\[ 2(-2)^3 - 5(-2)^2 - 17(-2) + 2 = -16 - 20 + 34 + 2 = 0 \][/tex]
So, [tex]\(x = -2\)[/tex] is a root.
Step 6: Perform polynomial division to factor out [tex]\((x + 2)\)[/tex]:
[tex]\[ 2x^3 - 5x^2 - 17x + 2 \div (x + 2) \][/tex]
Using synthetic division or long division, we find:
[tex]\[ 2x^3 - 5x^2 - 17x + 2 = (x + 2)(2x^2 - 9x - 1) \][/tex]
Step 7: Solve the quadratic equation [tex]\(2x^2 - 9x - 1 = 0\)[/tex] using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a = 2\)[/tex], [tex]\(b = -9\)[/tex], and [tex]\(c = -1\)[/tex].
[tex]\[ x = \frac{9 \pm \sqrt{81 + 8}}{4} = \frac{9 \pm \sqrt{89}}{4} \][/tex]
So, the two roots are:
[tex]\[ x = \frac{9 + \sqrt{89}}{4} \approx 2.85 \][/tex]
[tex]\[ x = \frac{9 - \sqrt{89}}{4} \approx -0.35 \][/tex]
Step 8: Combine all roots, including the factor obtained earlier:
The roots of the equation [tex]\(x^3 - 5x^2 + 2 = -x^3 + 17x\)[/tex] are:
[tex]\[ x = -2, \, 2.85, \, -0.35 \][/tex]
Rounded to the nearest hundredth, the non-integer roots are:
[tex]\[ x \approx 2.85, -0.35 \][/tex]
Thus, the roots of the polynomial equation are approximately:
[tex]\[ -2, 2.85, -0.35 \][/tex]
