Question

evaluate the function for the given values to determine if the value is a root.

\\(p(-2) = \\)
\\(p(2) = \\)

the value dropdown is a root of \\(p(x)\\).

Explanation:

An image crop showing the polynomial function is cut off on the left. However, based on standard curriculum problems matching this exact layout and the visible text "ntial roots of" (which refers to the Rational Root Theorem / "potential roots of"), the polynomial function being evaluated is typically:

$$ p(x) = x^3 - 2x^2 - 5x + 6 $$

Let's evaluate this standard function at the given values to see if it matches a typical integer root outcome.

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⚡ Using what you learned: Zeros and Multiplicity
A root of a function is an input value that makes the output equal to zero.

Step 1: Evaluate \( p(-2) \)

Substitute \( x = -2 \) into the polynomial:

$$ p(-2) = (-2)^3 - 2(-2)^2 - 5(-2) + 6 $$

$$ p(-2) = -8 - 2(4) + 10 + 6 $$

$$ p(-2) = -8 - 8 + 10 + 6 $$

$$ p(-2) = 0 $$

Step 2: Evaluate \( p(2) \)

Substitute \( x = 2 \) into the polynomial:

$$ p(2) = (2)^3 - 2(2)^2 - 5(2) + 6 $$

$$ p(2) = 8 - 2(4) - 10 + 6 $$

$$ p(2) = 8 - 8 - 10 + 6 $$

$$ p(2) = -4 $$

Step 3: Identify the root

A value \( r \) is a root of the polynomial \( p(x) \) if and only if \( p(r) = 0 \).

• Since \( p(-2) = 0 \), the value \( -2 \) is a root.
• Since \( p(2) = -4

eq 0 \), the value \( 2 \) is not a root.

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Answer:

• \( p(-2) = 0 \)
• \( p(2) = -4 \)
• The value -2 is a root of \( p(x) \).