We can certainly have polynomial functions as well as expressions. If \( f(x)=x^{3}-2 x^{2}+8 x-1 \) and \( g(x)=7 x^{2}-4 x+3 \) then find an expression for each of the following in standard form: \( \begin{array}{ll}\text { (a) } f(x)+g(x) & \text { (b) } f(x)-g(x)\end{array} \)

To find the expressions for \( f(x) + g(x) \) and \( f(x) - g(x) \) in standard form, we simply combine the terms of each polynomial. (a) For \( f(x) + g(x) \): \[ f(x) + g(x) = (x^{3} - 2x^{2} + 8x - 1) + (7x^{2} - 4x + 3) \] Combining like terms: \[ = x^{3} + (-2x^{2} + 7x^{2}) + (8x - 4x) + (-1 + 3) \] \[ = x^{3} + 5x^{2} + 4x + 2 \] So, the expression for \( f(x) + g(x) \) in standard form is: \[ f(x) + g(x) = x^{3} + 5x^{2} + 4x + 2 \] (b) For \( f(x) - g(x) \): \[ f(x) - g(x) = (x^{3} - 2x^{2} + 8x - 1) - (7x^{2} - 4x + 3) \] Distributing the negative sign: \[ = x^{3} - 2x^{2} + 8x - 1 - 7x^{2} + 4x - 3 \] Combining like terms: \[ = x^{3} + (-2x^{2} - 7x^{2}) + (8x + 4x) + (-1 - 3) \] \[ = x^{3} - 9x^{2} + 12x - 4 \] Thus, the expression for \( f(x) - g(x) \) in standard form is: \[ f(x) - g(x) = x^{3} - 9x^{2} + 12x - 4 \]