나머지정리와 인수정리

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)

$({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16)\div (x-2)$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)

$({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10)\div (x+5)$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)

$({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81)\div (x-3)$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)

$(4{x}^{4}-2{x}^{3}-4x+2)\div (2x-1)$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)

$(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2)\div (2x+1)$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x-2,\,4{x}^{3}-3{x}^{2}-8x+4$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x-2,\,3{x}^{4}-6{x}^{3}-5x+10$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x+3,\,-4{x}^{3}+5{x}^{2}+8$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x-2,\,4{x}^{4}-15{x}^{2}-4$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x-\frac{1}{2},\,2{x}^{4}-{x}^{3}+2x-1$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x+\frac{1}{3},\,3{x}^{4}+{x}^{3}-3x+1$

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is ${x}^{2}-x+3$

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is $({x}^{2}+2x+4)$

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is ${x}^{2}+2x+5$

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is ${x}^{2}+x+1$

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is ${x}^{2}+2x+2$

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{4{x}^{3}-33}{x-2}$

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{2{x}^{3}+25}{x+3}$

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{3{x}^{3}+2x-5}{x-1}$

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{-4{x}^{3}-{x}^{2}-12}{x+4}$