Pregunta
Find a polynomial of the specified degree that has the given zeros. Degree \( 3 ; \) zeros \( -2,2,4 \) \( P(x)=\square \)
Ask by Cole Marsh. in the United States
Feb 27,2025
Solución de inteligencia artificial de Upstudy
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A polynomial of degree 3 with zeros at -2, 2, and 4 is:
\( P(x) = x^3 - 4x^2 - 4x + 16 \)
Solución
A polynomial with zeros at –2, 2, and 4 can be constructed by writing the factors corresponding to these zeros. That is, if a number a is a zero, then (x – a) is a factor. So here, we have:
P(x) = (x + 2)(x – 2)(x – 4).
If you prefer to expand this product, start by multiplying the first two factors:
(x + 2)(x – 2) = x² – 4
Now multiply by the third factor:
P(x) = (x² – 4)(x – 4)
= x²·x – 4·x² – 4·x + 16
= x³ – 4x² – 4x + 16.
Thus, a valid degree 3 polynomial with the given zeros is:
P(x) = x³ – 4x² – 4x + 16.
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To construct a polynomial of degree 3 with the given zeros \(-2\), \(2\), and \(4\), we can use the fact that if \(r\) is a root, then \((x - r)\) is a factor of the polynomial. Therefore, the polynomial can be expressed as: \[ P(x) = (x + 2)(x - 2)(x - 4). \] Now, we can expand this expression: 1. First, multiply \((x + 2)\) and \((x - 2)\): \[ (x + 2)(x - 2) = x^2 - 4. \] 2. Next, multiply the result by \((x - 4)\): \[ P(x) = (x^2 - 4)(x - 4) = x^3 - 4x^2 - 4x + 16. \] Thus, the polynomial is: \[ P(x) = x^3 - 4x^2 - 4x + 16. \]
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