Answer :
To completely factor the polynomial [tex]\( x^3 + 5x^2 + 9x + 45 \)[/tex], follow these steps:
1. Identify a Common Factor: First, look for any common factors among all terms of the polynomial. Upon inspection, there isn't a common factor that can be factored out of all terms.
2. Use Synthetic Division or Factor Theorem: To factor the polynomial completely, start by testing possible rational roots (using the Rational Root Theorem, which tells us the potential roots might be the factors of the constant term). For this polynomial, let's test some potential simple roots, such as [tex]\( x = -5 \)[/tex].
3. Verification of a Root: You can manually substitute [tex]\( x = -5 \)[/tex] into the polynomial:
[tex]\[
(-5)^3 + 5(-5)^2 + 9(-5) + 45 = -125 + 125 - 45 + 45 = 0
\][/tex]
Since the result is zero, [tex]\( x = -5 \)[/tex] is indeed a root.
4. Divide the Polynomial: After finding a root, use polynomial division (synthetic division in case) to divide the original polynomial [tex]\( x^3 + 5x^2 + 9x + 45 \)[/tex] by [tex]\( x + 5 \)[/tex].
5. Perform the Division:
[tex]\[
\begin{array}{c|ccc}
-5 & 1 & 5 & 9 & 45 \\
& & -5 & -25 & -70 \\
\hline
& 1 & 0 & -16 & -25 \\
\end{array}
\][/tex]
This results in a quotient of [tex]\( x^2 + 0x + 9 = x^2 + 9 \)[/tex].
6. Finalize the Factoring: The polynomial [tex]\( x^3 + 5x^2 + 9x + 45 \)[/tex] can therefore be expressed as a product of its factors:
[tex]\[
x^3 + 5x^2 + 9x + 45 = (x + 5)(x^2 + 9)
\][/tex]
Thus, the completely factored form of the polynomial is [tex]\((x + 5)(x^2 + 9)\)[/tex].
1. Identify a Common Factor: First, look for any common factors among all terms of the polynomial. Upon inspection, there isn't a common factor that can be factored out of all terms.
2. Use Synthetic Division or Factor Theorem: To factor the polynomial completely, start by testing possible rational roots (using the Rational Root Theorem, which tells us the potential roots might be the factors of the constant term). For this polynomial, let's test some potential simple roots, such as [tex]\( x = -5 \)[/tex].
3. Verification of a Root: You can manually substitute [tex]\( x = -5 \)[/tex] into the polynomial:
[tex]\[
(-5)^3 + 5(-5)^2 + 9(-5) + 45 = -125 + 125 - 45 + 45 = 0
\][/tex]
Since the result is zero, [tex]\( x = -5 \)[/tex] is indeed a root.
4. Divide the Polynomial: After finding a root, use polynomial division (synthetic division in case) to divide the original polynomial [tex]\( x^3 + 5x^2 + 9x + 45 \)[/tex] by [tex]\( x + 5 \)[/tex].
5. Perform the Division:
[tex]\[
\begin{array}{c|ccc}
-5 & 1 & 5 & 9 & 45 \\
& & -5 & -25 & -70 \\
\hline
& 1 & 0 & -16 & -25 \\
\end{array}
\][/tex]
This results in a quotient of [tex]\( x^2 + 0x + 9 = x^2 + 9 \)[/tex].
6. Finalize the Factoring: The polynomial [tex]\( x^3 + 5x^2 + 9x + 45 \)[/tex] can therefore be expressed as a product of its factors:
[tex]\[
x^3 + 5x^2 + 9x + 45 = (x + 5)(x^2 + 9)
\][/tex]
Thus, the completely factored form of the polynomial is [tex]\((x + 5)(x^2 + 9)\)[/tex].
