Answer :
To factor the polynomial [tex]\( f(x) = x^4 - 2x^3 - 18x^2 + 6x + 45 \)[/tex] completely, follow these steps:
1. Identify Possible Rational Roots: According to the Rational Root Theorem, any rational root of the polynomial, when it exists, is a factor of the constant term (45) divided by a factor of the leading coefficient (1). Therefore, possible rational roots could include [tex]\( \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45 \)[/tex].
2. Test for Actual Roots: Check which of these potential roots actually make [tex]\( f(x) = 0 \)[/tex]. Through substitution, we find:
- [tex]\( f(5) = (5)^4 - 2(5)^3 - 18(5)^2 + 6(5) + 45 = 0 \)[/tex]
- [tex]\( f(-3) = (-3)^4 - 2(-3)^3 - 18(-3)^2 + 6(-3) + 45 = 0 \)[/tex]
Thus, [tex]\( x = 5 \)[/tex] and [tex]\( x = -3 \)[/tex] are roots of the polynomial.
3. Factor Out Linear Terms: Since [tex]\( x = 5 \)[/tex] and [tex]\( x = -3 \)[/tex] are roots, [tex]\( (x - 5) \)[/tex] and [tex]\( (x + 3) \)[/tex] are factors of the polynomial. Therefore, we can start factoring the polynomial as:
[tex]\[
f(x) = (x - 5)(x + 3)Q(x)
\][/tex]
where [tex]\( Q(x) \)[/tex] is a quadratic polynomial.
4. Divide the Polynomial by the Linear Factors: To find [tex]\( Q(x) \)[/tex], divide the original polynomial by [tex]\( (x - 5)(x + 3) \)[/tex].
5. Solve for the Quadratic Factor:
After performing the division, you will find:
[tex]\[
Q(x) = x^2 - 3
\][/tex]
6. Combine All Factors:
The completely factored form of the polynomial [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = (x - 5)(x + 3)(x^2 - 3)
\][/tex]
Now, the polynomial is successfully factored as [tex]\( (x - 5)(x + 3)(x^2 - 3) \)[/tex].
1. Identify Possible Rational Roots: According to the Rational Root Theorem, any rational root of the polynomial, when it exists, is a factor of the constant term (45) divided by a factor of the leading coefficient (1). Therefore, possible rational roots could include [tex]\( \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45 \)[/tex].
2. Test for Actual Roots: Check which of these potential roots actually make [tex]\( f(x) = 0 \)[/tex]. Through substitution, we find:
- [tex]\( f(5) = (5)^4 - 2(5)^3 - 18(5)^2 + 6(5) + 45 = 0 \)[/tex]
- [tex]\( f(-3) = (-3)^4 - 2(-3)^3 - 18(-3)^2 + 6(-3) + 45 = 0 \)[/tex]
Thus, [tex]\( x = 5 \)[/tex] and [tex]\( x = -3 \)[/tex] are roots of the polynomial.
3. Factor Out Linear Terms: Since [tex]\( x = 5 \)[/tex] and [tex]\( x = -3 \)[/tex] are roots, [tex]\( (x - 5) \)[/tex] and [tex]\( (x + 3) \)[/tex] are factors of the polynomial. Therefore, we can start factoring the polynomial as:
[tex]\[
f(x) = (x - 5)(x + 3)Q(x)
\][/tex]
where [tex]\( Q(x) \)[/tex] is a quadratic polynomial.
4. Divide the Polynomial by the Linear Factors: To find [tex]\( Q(x) \)[/tex], divide the original polynomial by [tex]\( (x - 5)(x + 3) \)[/tex].
5. Solve for the Quadratic Factor:
After performing the division, you will find:
[tex]\[
Q(x) = x^2 - 3
\][/tex]
6. Combine All Factors:
The completely factored form of the polynomial [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = (x - 5)(x + 3)(x^2 - 3)
\][/tex]
Now, the polynomial is successfully factored as [tex]\( (x - 5)(x + 3)(x^2 - 3) \)[/tex].
