The polynomial function
𝑓
(
𝑥
)
=
𝑥
3
−
14
𝑥
2
+
63
𝑥
−
90
f(x)=x
3
−14x
2
+63x−90​ has ​
𝑥
=
5
x=5​ as a root.
What are the remaining roots of
𝑓
(
𝑥
)
f(x)​​? Separate multiple answers with a comma.

Given the polynomial function \( f(x) = x^3 - 14x^2 + 63x - 90 \) and knowing that \( x = 5 \) is a root, we can find the remaining roots by performing polynomial division to factorize \( f(x) \).

1. **Polynomial Division**: Divide \( f(x) \) by \( (x - 5) \).

\[ f(x) = (x - 5) \cdot Q(x) \]

Perform the polynomial division:

\[ 
\begin{array}{r|rrr}
   & x^3 & -14x^2 & +63x & -90 \\
\hline
x-5 & x^2 & -9x & +18 \\
\end{array}
\]

The quotient \( Q(x) \) is found by dividing \( x^3 - 14x^2 + 63x - 90 \) by \( x - 5 \):

- \( x^3 \div x = x^2 \)
  - Multiply \( x^2 \) by \( x - 5 \) to get \( x^3 - 5x^2 \)
  - Subtract: \( (x^3 - 14x^2) - (x^3 - 5x^2) = -9x^2 \)

- \( -9x^2 \div x = -9x \)
  - Multiply \( -9x \) by \( x - 5 \) to get \( -9x^2 + 45x \)
  - Subtract: \( (-9x^2 + 63x) - (-9x^2 + 45x) = 18x \)

- \( 18x \div x = 18 \)
  - Multiply \( 18 \) by \( x - 5 \) to get \( 18x - 90 \)
  - Subtract: \( (18x - 90) - (18x - 90) = 0 \)

So, the quotient is \( Q(x) = x^2 - 9x + 18 \).

2. **Finding Remaining Roots**: Solve the quadratic equation \( x^2 - 9x + 18 = 0 \).

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1 \), \( b = -9 \), and \( c = 18 \).

\[ x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot 18}}{2 \cdot 1} \]
\[ x = \frac{9 \pm \sqrt{81 - 72}}{2} \]
\[ x = \frac{9 \pm \sqrt{9}}{2} \]
\[ x = \frac{9 \pm 3}{2} \]

Thus, the roots are:

\[ x = \frac{9 + 3}{2} = 6 \]
\[ x = \frac{9 - 3}{2} = 3 \]

Therefore, the remaining roots of \( f(x) \) are \( x = 6 \) and \( x = 3 \).

**Summary:**

The roots of the polynomial \( f(x) = x^3 - 14x^2 + 63x - 90 \) are \( 5, 6, \) and \( 3 \).

Would you like any further details or have any questions? Here are five more questions you might be interested in:

1. How do you perform synthetic division?
2. Can you explain the Remainder Theorem and its applications?
3. What is the Factor Theorem and how is it used?
4. How do you find complex roots of a polynomial?
5. Can you explain how to graph polynomial functions?

**Tip:** When solving polynomial equations, always check for possible rational roots using the Rational Root Theorem before proceeding with polynomial division.

Learn how to find the remaining roots of the polynomial function f(x) = x^3 - 14x^2 + 63x - 90 given that x = 5 is a root. Detailed step-by-step solution involving polynomial division, quadratic equation solving, and application of the Remainder Theorem and Factor Theorem. Suitable for high school students.

Math Problem Statement

The polynomial function 𝑓 ( 𝑥 )

𝑥 3 − 14 𝑥 2 + 63 𝑥 − 90 f(x)=x 3 −14x 2 +63x−90 has 𝑥

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