Answer :
To write the polynomial [tex]\(x^4 + x^3 - 19x^2 + 11x + 30\)[/tex] as a product of linear factors, we will find its roots and express the polynomial as a product involving these roots. Here is a step-by-step guide:
1. Identify the Polynomial:
We start with the polynomial [tex]\(f(x) = x^4 + x^3 - 19x^2 + 11x + 30\)[/tex].
2. Finding Roots:
To express the polynomial as a product of linear factors, we need to determine the roots of the polynomial. These roots are the values of [tex]\(x\)[/tex] that satisfy [tex]\(f(x) = 0\)[/tex].
3. Trial and Error or Use of Mathematical Tools:
Begin by testing some possible rational roots, like factors of the constant term, 30. Through either manual testing or using a polynomial factorization technique, we find that the roots are [tex]\(x = 3\)[/tex], [tex]\(x = 2\)[/tex], [tex]\(x = -1\)[/tex], and [tex]\(x = -5\)[/tex].
4. Forming Linear Factors:
With the roots identified, each root leads to a linear factor. If [tex]\(r\)[/tex] is a root, then [tex]\((x - r)\)[/tex] is a factor. Therefore, the linear factors for the identified roots are:
- [tex]\(x - 3\)[/tex]
- [tex]\(x - 2\)[/tex]
- [tex]\(x + 1\)[/tex]
- [tex]\(x + 5\)[/tex]
5. Expressing the Polynomial as a Product of Linear Factors:
Using the roots, we can express the original polynomial as follows:
[tex]\[
(x - 3)(x - 2)(x + 1)(x + 5)
\][/tex]
By confirming these steps, we conclude that the polynomial [tex]\(x^4 + x^3 - 19x^2 + 11x + 30\)[/tex] can be factored into the linear factors [tex]\((x - 3)(x - 2)(x + 1)(x + 5)\)[/tex].
1. Identify the Polynomial:
We start with the polynomial [tex]\(f(x) = x^4 + x^3 - 19x^2 + 11x + 30\)[/tex].
2. Finding Roots:
To express the polynomial as a product of linear factors, we need to determine the roots of the polynomial. These roots are the values of [tex]\(x\)[/tex] that satisfy [tex]\(f(x) = 0\)[/tex].
3. Trial and Error or Use of Mathematical Tools:
Begin by testing some possible rational roots, like factors of the constant term, 30. Through either manual testing or using a polynomial factorization technique, we find that the roots are [tex]\(x = 3\)[/tex], [tex]\(x = 2\)[/tex], [tex]\(x = -1\)[/tex], and [tex]\(x = -5\)[/tex].
4. Forming Linear Factors:
With the roots identified, each root leads to a linear factor. If [tex]\(r\)[/tex] is a root, then [tex]\((x - r)\)[/tex] is a factor. Therefore, the linear factors for the identified roots are:
- [tex]\(x - 3\)[/tex]
- [tex]\(x - 2\)[/tex]
- [tex]\(x + 1\)[/tex]
- [tex]\(x + 5\)[/tex]
5. Expressing the Polynomial as a Product of Linear Factors:
Using the roots, we can express the original polynomial as follows:
[tex]\[
(x - 3)(x - 2)(x + 1)(x + 5)
\][/tex]
By confirming these steps, we conclude that the polynomial [tex]\(x^4 + x^3 - 19x^2 + 11x + 30\)[/tex] can be factored into the linear factors [tex]\((x - 3)(x - 2)(x + 1)(x + 5)\)[/tex].
