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Simplify : \( \sqrt{63}-\sqrt{175}+\sqrt{28} \) If \( p(x)=4 x^{3}-6 x^{2}+3 x-14 \), then find the value of \( p(-1) \) and \( p(0) \)

Ask by Hill May. in India
Dec 31,2024

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Simplify \( \sqrt{63} - \sqrt{175} + \sqrt{28} \) to get 0. For the polynomial \( p(x) = 4x^3 - 6x^2 + 3x - 14 \): - \( p(-1) = -27 \) - \( p(0) = -14 \)

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To simplify \( \sqrt{63}-\sqrt{175}+\sqrt{28} \), we first break down each square root into its prime factors: 1. \( \sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7} \) 2. \( \sqrt{175} = \sqrt{25 \cdot 7} = 5\sqrt{7} \) 3. \( \sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7} \) Now substituting these back into the expression: \[ \sqrt{63}-\sqrt{175}+\sqrt{28} = 3\sqrt{7} - 5\sqrt{7} + 2\sqrt{7} \] Combining like terms gives: \[ (3 - 5 + 2)\sqrt{7} = 0\sqrt{7} = 0 \] So, the simplified expression is \( 0 \). Next, let's find \( p(-1) \) and \( p(0) \) for the polynomial \( p(x) = 4x^{3} - 6x^{2} + 3x - 14 \). Calculating \( p(-1) \): \[ p(-1) = 4(-1)^{3} - 6(-1)^{2} + 3(-1) - 14 \] \[ = 4(-1) - 6(1) - 3 - 14 \] \[ = -4 - 6 - 3 - 14 = -27 \] Thus, \( p(-1) = -27 \). Now calculating \( p(0) \): \[ p(0) = 4(0)^{3} - 6(0)^{2} + 3(0) - 14 \] \[ = 0 - 0 + 0 - 14 = -14 \] Thus, \( p(0) = -14 \). In summary, the results are \( p(-1) = -27 \) and \( p(0) = -14 \).

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