The polynomial function y = x3 + 8×2 – 12x – 144 has a root located at (-6,0).
a) Use polynomial division AND factoring to write the polynomial in fully factored form. Show all work that supports your answer. Fully factored form: y
b) Identify the degree and all roots of the original polynomial function. *Write roots as (x, y) points! Degree Crossing Root(s) Bouncing Root(s)
The correct answer and explanation is :
Let’s solve the given polynomial function step by step.
Given Polynomial:
[
y = x^3 + 8x^2 – 12x – 144
]
We are given that ( (-6,0) ) is a root, which means ( x+6 ) is a factor of the polynomial.
Step 1: Polynomial Division
We divide ( x^3 + 8x^2 – 12x – 144 ) by ( x + 6 ) using synthetic division.
Synthetic Division:
Set up the division with ( x+6 ):
-6 | 1 8 -12 -144
| -6 -12 144
-----------------
1 2 -24 0The quotient is ( x^2 + 2x – 24 ), and the remainder is 0, confirming that ( x+6 ) is a factor.
Step 2: Factoring the Quotient
Now, we factor ( x^2 + 2x – 24 ).
We need two numbers that multiply to -24 and add to 2. These numbers are 6 and -4.
[
x^2 + 2x – 24 = (x + 6)(x – 4)
]
Step 3: Fully Factored Form
Since we already divided by ( x+6 ), the fully factored form of the polynomial is:
[
y = (x + 6)(x + 6)(x – 4)
]
or written with exponents:
[
y = (x + 6)^2 (x – 4)
]
Part B: Degree and Roots
- Degree: The highest exponent in the polynomial is 3, so the degree is 3.
- Roots (x, y) points:
- The factor ( (x+6)^2 ) tells us that x = -6 is a repeated root (bounces at x = -6).
- The factor ( (x-4) ) tells us that x = 4 is a crossing root. So, the roots are:
- Bouncing Root: ( (-6, 0) ) (because ( (x+6)^2 ) means it bounces at x = -6)
- Crossing Root: ( (4, 0) ) (since ( (x-4) ) appears only once)
Final Answer:
- Fully Factored Form: ( y = (x+6)^2 (x-4) )
- Degree: ( 3 )
- Roots: ( (-6,0) ), ( (-6,0) ), ( (4,0) )
- Bouncing Root: ( (-6,0) )
- Crossing Root: ( (4,0) )
Explanation (300 Words)
The given polynomial function is ( y = x^3 + 8x^2 – 12x – 144 ), and we are given that one of its roots is ( (-6,0) ). This means ( x+6 ) is a factor of the polynomial.
To find the fully factored form, we perform polynomial division. Using synthetic division, we divide ( x^3 + 8x^2 – 12x – 144 ) by ( x+6 ), which gives the quotient ( x^2 + 2x – 24 ) with a remainder of 0, confirming ( x+6 ) as a factor.
Next, we factor ( x^2 + 2x – 24 ). Finding two numbers that multiply to -24 and add to 2, we get 6 and -4. So, it factors as ( (x+6)(x-4) ). Since we already had an ( x+6 ) factor from the division, the fully factored form of the polynomial is:
[
y = (x+6)^2 (x-4)
]
This form helps us identify the roots. The exponent 2 on ( (x+6) ) means the root ( x = -6 ) has even multiplicity, so the graph of the function bounces at ( x = -6 ). The root ( x = 4 ) has an exponent of 1, meaning it crosses the x-axis at ( x = 4 ).
Thus, the function has a degree of 3 (highest power of x), a bouncing root at (-6,0), and a crossing root at (4,0).
This solution shows how polynomial division and factoring help break down the function into its simplest form and understand its behavior graphically.