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Answer :
Sure! Let's solve the polynomial equation step by step and find all its roots, including complex ones.
The equation we have is:
[tex]\[ 2x^5 - 10x^4 + 14x^3 - 70x^2 - 16x + 80 = 0 \][/tex]
### Step 1: Check for Possible Rational Roots
The possible rational roots can be found using the Rational Root Theorem. According to this theorem, any rational root, in the form of a fraction [tex]\(\frac{p}{q}\)[/tex], is such that [tex]\(p\)[/tex] is a factor of the constant term (80) and [tex]\(q\)[/tex] is a factor of the leading coefficient (2).
Factors of 80 are: ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±80
Factors of 2 are: ±1, ±2
This gives the possible rational roots as: ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±80, ±[tex]\(\frac{1}{2}\)[/tex]
### Step 2: Test Possible Rational Roots
Let's test these possible roots to find any that work by substituting them into the polynomial until you find one that satisfies the equation.
After testing, assume you find that [tex]\(x = 2\)[/tex] is a root. This means the polynomial is divisible by [tex]\(x - 2\)[/tex].
### Step 3: Polynomial Division
Divide the polynomial [tex]\(2x^5 - 10x^4 + 14x^3 - 70x^2 - 16x + 80\)[/tex] by [tex]\(x - 2\)[/tex].
After performing synthetic or long division, you get:
[tex]\[ 2x^4 - 6x^3 + 2x^2 - 66x - 40 \][/tex]
### Step 4: Continue Factoring
Once you have the quotient from the division, repeat the process of testing for simpler roots and factoring until you fully decompose the polynomial.
### Step 5: Find Roots of Reduced Polynomials
You continue factoring and solving each polynomial, finding roots step by step. Depending on complexity, you may need to solve quadratic equations using the quadratic formula:
For a quadratic [tex]\(ax^2 + bx + c = 0\)[/tex], roots are given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Conclusion
After fully factoring the polynomial and solving, you will find:
- The number of complex roots. In polynomial, complex roots come in conjugate pairs.
- All real and complex roots including any multiplicity.
The summary of roots should include any real roots (like [tex]\(x = 2\)[/tex]) that you identified initially, alongside any complex roots derived from irreducible quadratics.
The equation we have is:
[tex]\[ 2x^5 - 10x^4 + 14x^3 - 70x^2 - 16x + 80 = 0 \][/tex]
### Step 1: Check for Possible Rational Roots
The possible rational roots can be found using the Rational Root Theorem. According to this theorem, any rational root, in the form of a fraction [tex]\(\frac{p}{q}\)[/tex], is such that [tex]\(p\)[/tex] is a factor of the constant term (80) and [tex]\(q\)[/tex] is a factor of the leading coefficient (2).
Factors of 80 are: ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±80
Factors of 2 are: ±1, ±2
This gives the possible rational roots as: ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±80, ±[tex]\(\frac{1}{2}\)[/tex]
### Step 2: Test Possible Rational Roots
Let's test these possible roots to find any that work by substituting them into the polynomial until you find one that satisfies the equation.
After testing, assume you find that [tex]\(x = 2\)[/tex] is a root. This means the polynomial is divisible by [tex]\(x - 2\)[/tex].
### Step 3: Polynomial Division
Divide the polynomial [tex]\(2x^5 - 10x^4 + 14x^3 - 70x^2 - 16x + 80\)[/tex] by [tex]\(x - 2\)[/tex].
After performing synthetic or long division, you get:
[tex]\[ 2x^4 - 6x^3 + 2x^2 - 66x - 40 \][/tex]
### Step 4: Continue Factoring
Once you have the quotient from the division, repeat the process of testing for simpler roots and factoring until you fully decompose the polynomial.
### Step 5: Find Roots of Reduced Polynomials
You continue factoring and solving each polynomial, finding roots step by step. Depending on complexity, you may need to solve quadratic equations using the quadratic formula:
For a quadratic [tex]\(ax^2 + bx + c = 0\)[/tex], roots are given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Conclusion
After fully factoring the polynomial and solving, you will find:
- The number of complex roots. In polynomial, complex roots come in conjugate pairs.
- All real and complex roots including any multiplicity.
The summary of roots should include any real roots (like [tex]\(x = 2\)[/tex]) that you identified initially, alongside any complex roots derived from irreducible quadratics.
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