Let \( f(x)=7 x^{2}-3 x+4 \). When \( f(x) \) is divided by \( x+2 \) the remainder is: When \( f(x) \) is divided by \( x+1 \) the remainder is: When \( f(x) \) is divided by \( x \) the remainder is: When \( f(x) \) is divided by \( x-1 \) the remainder is: When \( f(x) \) is divided by \( x-2 \) the remainder is:

To find the remainders when the polynomial \( f(x) = 7x^2 - 3x + 4 \) is divided by \( x + a \), we can use the Remainder Theorem. This theorem states that the remainder of the division of a polynomial \( f(x) \) by \( x - a \) is equal to \( f(a) \). 1. For \( x + 2 \) (which is \( x - (-2) \)): \[ f(-2) = 7(-2)^2 - 3(-2) + 4 = 7(4) + 6 + 4 = 28 + 6 + 4 = 38 \] 2. For \( x + 1 \) (which is \( x - (-1) \)): \[ f(-1) = 7(-1)^2 - 3(-1) + 4 = 7(1) + 3 + 4 = 7 + 3 + 4 = 14 \] 3. For \( x \) (which is \( x - 0 \)): \[ f(0) = 7(0)^2 - 3(0) + 4 = 0 + 0 + 4 = 4 \] 4. For \( x - 1 \): \[ f(1) = 7(1)^2 - 3(1) + 4 = 7(1) - 3 + 4 = 7 - 3 + 4 = 8 \] 5. For \( x - 2 \): \[ f(2) = 7(2)^2 - 3(2) + 4 = 7(4) - 6 + 4 = 28 - 6 + 4 = 26 \] To summarize: - When \( f(x) \) is divided by \( x + 2 \), the remainder is **38**. - When \( f(x) \) is divided by \( x + 1 \), the remainder is **14**. - When \( f(x) \) is divided by \( x \), the remainder is **4**. - When \( f(x) \) is divided by \( x - 1 \), the remainder is **8**. - When \( f(x) \) is divided by \( x - 2 \), the remainder is **26**.