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. According to this theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \). 1. **Remainder when divided by \( x + 2 \):** Here, we use \( c = -2 \): \[ f(-2) = 7(-2)^2 - 3(-2) + 4 = 7(4) + 6 + 4 = 28 + 6 + 4 = 38 \] So, the remainder is **38**. 2. **Remainder when divided by \( x + 1 \):** Here, use \( c = -1 \): \[ f(-1) = 7(-1)^2 - 3(-1) + 4 = 7(1) + 3 + 4 = 7 + 3 + 4 = 14 \] So, the remainder is **14**. 3. **Remainder when divided by \( x \):** Here, use \( c = 0 \): \[ f(0) = 7(0)^2 - 3(0) + 4 = 0 + 0 + 4 = 4 \] So, the remainder is **4**. 4. **Remainder when divided by \( x - 1 \):** Here, use \( c = 1 \): \[ f(1) = 7(1)^2 - 3(1) + 4 = 7(1) - 3 + 4 = 7 - 3 + 4 = 8 \] So, the remainder is **8**. 5. **Remainder when divided by \( x - 2 \):** Here, use \( c = 2 \): \[ f(2) = 7(2)^2 - 3(2) + 4 = 7(4) - 6 + 4 = 28 - 6 + 4 = 26 \] So, the remainder is **26**. In summary: - Remainder when divided by \( x + 2 \): **38** - Remainder when divided by \( x + 1 \): **14** - Remainder when divided by \( x \): **4** - Remainder when divided by \( x - 1 \): **8** - Remainder when divided by \( x - 2 \): **26**