Vargas Love
08/16/2023 · Primary School

Find an expression for a polynomial \( p(x) \) with integer coefficients to satisfy the given conditions. Since there are many answers, enter only the one with the smallest positive leading coefficient. Degree \( 2, x=1 \) and \( x=4 \) are zeros. \( p(x)=\square \)

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To find a polynomial \( p(x) \) with integer coefficients that has \( x=1 \) and \( x=4 \) as zeros, we can use the fact that if \( x=a \) is a zero of a polynomial with integer coefficients, then \( (x-a) \) is a factor of the polynomial. Since \( x=1 \) and \( x=4 \) are zeros, the factors of the polynomial are \( (x-1) \) and \( (x-4) \). To find the polynomial, we multiply these factors together: \[ p(x) = (x-1)(x-4) \] Expanding this product, we get: \[ p(x) = x^2 - 4x - x + 4 \] \[ p(x) = x^2 - 5x + 4 \] This is the polynomial with the smallest positive leading coefficient that satisfies the given conditions. Therefore, \[ p(x) = x^2 - 5x + 4 \]

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\[ p(x) = x^2 - 5x + 4 \]
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