To factorize the cubic polynomial \( x^3 - 9x^2 + 23x - 15 \), we will follow a systematic approach. Here are the steps:
Step 1: Identify possible rational roots
The Rational Root Theorem states that any rational root of the polynomial \( P(x) = x^3 - 9x^2 + 23x - 15 \) must be a factor of the constant term (\(-15\)) divided by a factor of the leading coefficient (which is \(1\)). Therefore, the possible rational roots are:
\[
\pm 1, \pm 3, \pm 5, \pm 15
\]
Step 2: Test the possible roots
We will test these possible roots by substituting them into the polynomial \( P(x) \).
#### Test \( x = 1 \):
\[
P(1) = 1^3 - 9(1)^2 + 23(1) - 15 = 1 - 9 + 23 - 15 = 0
\]
Since \( P(1) = 0 \), \( x = 1 \) is a root. Therefore, \( (x - 1) \) is a factor of the polynomial.
Step 3: Perform polynomial division
We will divide \( x^3 - 9x^2 + 23x - 15 \) by \( x - 1 \) using synthetic division.
#### Synthetic division for \( x - 1 \):
\[
\begin{array}{r|rrrr}
1 & 1 & -9 & 23 & -15 \\
& & 1 & -8 & 15 \\
\hline
& 1 & -8 & 15 & 0 \\
\end{array}
\]
The quotient is \( x^2 - 8x + 15 \), and the remainder is \( 0 \). Therefore:
\[
x^3 - 9x^2 + 23x - 15 = (x - 1)(x^2 - 8x + 15)
\]
Step 4: Factorize the quadratic polynomial
Next, we factorize the quadratic polynomial \( x^2 - 8x + 15 \). We look for two numbers that multiply to \( 15 \) (the constant term) and add up to \( -8 \) (the coefficient of \( x \)).
The numbers are \( -3 \) and \( -5 \) because:
\[
-3 \cdot -5 = 15 \quad \text{and} \quad -3 + (-5) = -8
\]
Thus, we can write:
\[
x^2 - 8x + 15 = (x - 3)(x - 5)
\]
Step 5: Write the complete factorization
Combining all the factors, we have:
\[
x^3 - 9x^2 + 23x - 15 = (x - 1)(x - 3)(x - 5)
\]
Final Answer:
\[
\boxed{(x - 1)(x - 3)(x - 5)}
\]
Parent Tip: Review the logic above to help your child master the concept of factoring cubic polynomials worksheet.
