1. **Problem statement:** Use the Remainder Theorem to find the remainder when each polynomial is divided by the given divisor or evaluated at the given value of $x$. 2. **Remainder Theorem:** For a polynomial $f(x)$, the remainder when divided by $(x - a)$ is $f(a)$. --- ### Problem 1: Find remainder of $6x^{456} + 2$ when divided by $(x - 1)$ 3. Substitute $x = 1$ into the polynomial: $$6(1)^{456} + 2 = 6 \times 1 + 2 = 6 + 2 = 8$$ **Remainder:** $8$ --- ### Problem 2: Find remainder of $6x^{500} - 4x^{400} - 2x^{300} + 5x^{200} - 14x + 5$ at $x = -1$ 4. Substitute $x = -1$: $$6(-1)^{500} - 4(-1)^{400} - 2(-1)^{300} + 5(-1)^{200} - 14(-1) + 5$$ Calculate powers: - $(-1)^{500} = 1$ - $(-1)^{400} = 1$ - $(-1)^{300} = -1$ - $(-1)^{200} = 1$ So: $$6(1) - 4(1) - 2(-1) + 5(1) + 14 + 5 = 6 - 4 + 2 + 5 + 14 + 5$$ Sum: $$6 - 4 = 2$$ $$2 + 2 = 4$$ $$4 + 5 = 9$$ $$9 + 14 = 23$$ $$23 + 5 = 28$$ **Remainder:** $28$ --- ### Problem 3: Find remainder of $x^3 - kx - 11x + 12$ at $x = 4$ 5. Substitute $x = 4$: $$4^3 - k(4) - 11(4) + 12 = 64 - 4k - 44 + 12$$ Simplify constants: $$64 - 44 + 12 = 32$$ So remainder is: $$32 - 4k$$ **Remainder:** $32 - 4k$ --- ### Problem 4: Find remainder of $3x^3 + 10^2 - 27k - 10$ at $x = 2$ 6. Substitute $x = 2$: $$3(2)^3 + 10^2 - 27k - 10 = 3(8) + 100 - 27k - 10$$ Simplify constants: $$24 + 100 - 10 = 114$$ So remainder is: $$114 - 27k$$ **Remainder:** $114 - 27k$