go
topic: finding the roots and factors of a polynomial
use the given root to find the remaining roots. then write the function in factored form.
14.
$f(x) = x^3 - x^2 - 17x - 15$
$x = -1$
15.
$f(x) = x^3 - 3x^2 - 61x + 63$
$x = 1$
16.
$f(x) = 6x^3 - 18x^2 - 60x$
$x = 0$
17.
$f(x) = x^3 - 14x^2 + 57x - 72$
$x = 8$
18. a relationship exists between the roots of a function and the constant term of the function. look back at the roots and the constant term in each problem. make a statement about anything you notice.

Question

Accepted Answer

### Problem 14:
# Explanation:
## Step1: Divide by (x + 1)
Since $ x = -1 $ is a root, $ (x + 1) $ is a factor. Use polynomial long division or synthetic division. Let's use synthetic division:
$$
\begin{array}{r|rrrr}
-1 & 1 & -1 & -17 & -15 \
 & & -1 & 2 & 15 \
\hline
 & 1 & -2 & -15 & 0 \
\end{array}
$$
The quotient is $ x^2 - 2x - 15 $.

## Step2: Factor the quadratic
Factor $ x^2 - 2x - 15 $: $ x^2 - 2x - 15 = (x - 5)(x + 3) $.

## Step3: Find roots and factored form
Roots are $ x = -1, 5, -3 $. Factored form: $ f(x) = (x + 1)(x - 5)(x + 3) $.

# Answer:
Remaining roots: $ 5, -3 $; Factored form: $ f(x) = (x + 1)(x - 5)(x + 3) $

### Problem 15:
# Explanation:
## Step1: Divide by (x - 1)
Since $ x = 1 $ is a root, $ (x - 1) $ is a factor. Synthetic division:
$$
\begin{array}{r|rrrr}
1 & 1 & -3 & -61 & 63 \
 & & 1 & -2 & -63 \
\hline
 & 1 & -2 & -63 & 0 \
\end{array}
$$
Quotient: $ x^2 - 2x - 63 $.

## Step2: Factor the quadratic
Factor $ x^2 - 2x - 63 $: $ x^2 - 2x - 63 = (x - 9)(x + 7) $.

## Step3: Find roots and factored form
Roots: $ 1, 9, -7 $. Factored form: $ f(x) = (x - 1)(x - 9)(x + 7) $.

# Answer:
Remaining roots: $ 9, -7 $; Factored form: $ f(x) = (x - 1)(x - 9)(x + 7) $

### Problem 16:
# Explanation:
## Step1: Factor out x
Given $ x = 0 $ is a root, factor out $ x $: $ f(x) = x(6x^2 - 18x - 60) $.

## Step2: Simplify the quadratic
Divide quadratic by 6: $ 6x^2 - 18x - 60 = 6(x^2 - 3x - 10) $.

## Step3: Factor the quadratic
Factor $ x^2 - 3x - 10 $: $ x^2 - 3x - 10 = (x - 5)(x + 2) $.

## Step4: Find roots and factored form
Roots: $ x = 0, 5, -2 $. Factored form: $ f(x) = 6x(x - 5)(x + 2) $.

# Answer:
Remaining roots: $ 5, -2 $; Factored form: $ f(x) = 6x(x - 5)(x + 2) $

### Problem 17:
# Explanation:
## Step1: Divide by (x - 8)
Since $ x = 8 $ is a root, $ (x - 8) $ is a factor. Synthetic division:
$$
\begin{array}{r|rrrr}
8 & 1 & -14 & 57 & -72 \
 & & 8 & -48 & 72 \
\hline
 & 1 & -6 & 9 & 0 \
\end{array}
$$
Quotient: $ x^2 - 6x + 9 $.

## Step2: Factor the quadratic
Factor $ x^2 - 6x + 9 $: $ x^2 - 6x + 9 = (x - 3)^2 $.

## Step3: Find roots and factored form
Roots: $ x = 8, 3, 3 $. Factored form: $ f(x) = (x - 8)(x - 3)^2 $.

# Answer:
Remaining root: $ 3 $ (double root); Factored form: $ f(x) = (x - 8)(x - 3)^2 $

### Problem 18:
# Brief Explanations:
For a cubic polynomial $ f(x) = ax^3 + bx^2 + cx + d $ (with $ a = 1 $ for monic polynomials), the product of the roots (considering multiplicity) is $ -d $ (by Vieta's formulas). For example, in problem 14: roots $ -1, 5, -3 $, product is $ (-1)(5)(-3) = 15 $, and $ d = -15 $, so $ -d = 15 $. In problem 15: roots $ 1, 9, -7 $, product is $ (1)(9)(-7) = -63 $, and $ d = 63 $, so $ -d = -63 $. For polynomials with $ a
eq 1 $, adjust for the leading coefficient. For example, problem 16: $ f(x) = 6x^3 - 18x^2 - 60x $, roots $ 0, 5, -2 $, product is $ 0 \times 5 \times (-2) = 0 $, and $ d = 0 $. In problem 17: roots $ 8, 3, 3 $, product is $ 8 \times 3 \times 3 = 72 $, and $ d = -72 $, so $ -d = 72 $. The pattern is that for a cubic polynomial $ f(x) = ax^3 + bx^2 + cx + d $, the product of the roots (counting multiplicity) is $ -\frac{d}{a} $.

# Answer:
For a cubic polynomial $ f(x) = ax^3 + bx^2 + cx + d $, the product of the roots (with multiplicity) is $ -\frac{d}{a} $. For monic cubics ($ a = 1 $), the product of the roots is $ -d $.

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QUESTION IMAGE

Question

Explanation:

Problem 14:

Step1: Divide by (x + 1)

Step2: Factor the quadratic

Step3: Find roots and factored form

Step1: Divide by (x - 1)

Step2: Factor the quadratic

Step3: Find roots and factored form

Step1: Factor out x

Step2: Simplify the quadratic

Step3: Factor the quadratic

Step4: Find roots and factored form

Answer:

Problem 15: