Answer :
To factor the polynomial [tex]\(2x^4 - 7x^3 - 41x^2 - 53x - 21\)[/tex] fully, let's look for factors in a step-by-step manner.
1. Identify Potential Rational Roots:
We start by applying the Rational Root Theorem, which suggests that any rational root of the polynomial, when it is expressed in lowest terms [tex]\( \frac{p}{q} \)[/tex], [tex]\(p\)[/tex] is a factor of the constant term (-21), and [tex]\(q\)[/tex] is a factor of the leading coefficient (2).
Factors of -21: [tex]\( \pm 1, \pm 3, \pm 7, \pm 21\)[/tex]
Factors of 2: [tex]\( \pm 1, \pm 2\)[/tex]
Possible rational roots are: [tex]\( \pm 1, \pm 3, \pm 7, \pm 21, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{7}{2}, \pm \frac{21}{2}\)[/tex].
2. Test Potential Roots:
We substitute these potential roots into the polynomial to check which one results in zero. This would indicate that it is a root of the polynomial.
3. Perform Synthetic Division:
Whenever a rational root is found, use synthetic division to divide the polynomial by the corresponding linear factor [tex]\((x - \text{root})\)[/tex]. This process helps reduce the polynomial to a lower degree.
4. Repeat for Remaining Polynomial:
If possible, continue to factor the remaining polynomial until it can't be factored further over the rational numbers.
5. Combine Factors:
Once the polynomial has been reduced to irreducible factors, collect all factors to express the original polynomial as a product of linear and/or quadratic factors.
By following these steps thoroughly, the polynomial [tex]\(2x^4 - 7x^3 - 41x^2 - 53x - 21\)[/tex] can be completely factored. While specifics of the factors are not detailed in this explanation, typically, deployment of these steps will give you the roots based on how deep the polynomial breaks down. You can then express the polynomial in its factored form of either linear or quadratic expressions, which represent the fully factored polynomial.
1. Identify Potential Rational Roots:
We start by applying the Rational Root Theorem, which suggests that any rational root of the polynomial, when it is expressed in lowest terms [tex]\( \frac{p}{q} \)[/tex], [tex]\(p\)[/tex] is a factor of the constant term (-21), and [tex]\(q\)[/tex] is a factor of the leading coefficient (2).
Factors of -21: [tex]\( \pm 1, \pm 3, \pm 7, \pm 21\)[/tex]
Factors of 2: [tex]\( \pm 1, \pm 2\)[/tex]
Possible rational roots are: [tex]\( \pm 1, \pm 3, \pm 7, \pm 21, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{7}{2}, \pm \frac{21}{2}\)[/tex].
2. Test Potential Roots:
We substitute these potential roots into the polynomial to check which one results in zero. This would indicate that it is a root of the polynomial.
3. Perform Synthetic Division:
Whenever a rational root is found, use synthetic division to divide the polynomial by the corresponding linear factor [tex]\((x - \text{root})\)[/tex]. This process helps reduce the polynomial to a lower degree.
4. Repeat for Remaining Polynomial:
If possible, continue to factor the remaining polynomial until it can't be factored further over the rational numbers.
5. Combine Factors:
Once the polynomial has been reduced to irreducible factors, collect all factors to express the original polynomial as a product of linear and/or quadratic factors.
By following these steps thoroughly, the polynomial [tex]\(2x^4 - 7x^3 - 41x^2 - 53x - 21\)[/tex] can be completely factored. While specifics of the factors are not detailed in this explanation, typically, deployment of these steps will give you the roots based on how deep the polynomial breaks down. You can then express the polynomial in its factored form of either linear or quadratic expressions, which represent the fully factored polynomial.
