Question

one factor of \\(f(x) = 5x^3 + 5x^2 - 170x + 280\\) is \\((x + 7)\\). what are all the roots of the function? use the remainder theorem.

\\(x = -4, x = -2, \text{ or } x = 7\\)
\\(x = -7, x = 2, \text{ or } x = 4\\)
\\(x = -7, x = 5, \text{ or } x = 280\\)
\\(x = -280, x = -5, \text{ or } x = 7\\)

Explanation:

🆕 New Concept Discovered: Polynomial Division and Factor Theorem
Finding all roots using a known linear factor.

Step 1: Identify the first root

Since \( (x + 7) \) is a factor of the polynomial \( f(x) = 5x^3 + 5x^2 - 170x + 280 \), we set the factor to zero to find the first root:

$$ x + 7 = 0 \implies x = -7 $$

This means \( x = -7 \) is one of the roots. Looking at the multiple-choice options, only the first two options contain \( x = -7 \):

• Option 1: \( x = -4, x = -2, \text{ or } x = 7 \) (does not contain \( -7 \))
• Option 2: \( x = -7, x = 2, \text{ or } x = 4 \) (contains \( -7 \))
• Option 3: \( x = -7, x = 5, \text{ or } x = 280 \) (contains \( -7 \))

Step 2: Divide the polynomial by the known factor

We divide \( f(x) = 5x^3 + 5x^2 - 170x + 280 \) by \( (x + 7) \) using synthetic division with \( c = -7 \):

$$
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$$

The quotient is the quadratic polynomial:

$$ q(x) = 5x^2 - 30x + 40 $$

Step 3: Find the remaining roots

Set the quotient to zero to find the remaining roots:

$$ 5x^2 - 30x + 40 = 0 $$

Divide the entire equation by \( 5 \):

$$ x^2 - 6x + 8 = 0 $$

Factor the quadratic equation:

$$ (x - 2)(x - 4) = 0 $$

This gives the remaining roots:

$$ x = 2 \quad \text{and} \quad x = 4 $$

Combining all roots, we have:

$$ x = -7, \quad x = 2, \quad \text{or} \quad x = 4 $$

Answer:

\( x = -7, x = 2, \text{ or } x = 4 \)