다항식 전체 문제

For the following exercises, use the graphs to write a polynomial function of least degree.

For the following exercises, write the polynomial function that models the given situation.

A rectangle has a length of 10 units and a width of 8 units. Squares of $x$ by $x$ units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of $x.$

For the following exercises, write the polynomial function that models the given situation.

Consider the same rectangle of the preceding problem. Squares of $2x$ by $2x$ units are cut out of each corner. Express the volume of the box as a polynomial in terms of $x.$

For the following exercises, write the polynomial function that models the given situation.

A square has sides of 12 units. Squares $x+1$ by $x+1$ units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of $x.$

For the following exercises, write the polynomial function that models the given situation.

A cylinder has a radius of $x+2$ units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.

For the following exercises, write the polynomial function that models the given situation.

A right circular cone has a radius of $3x+6$ and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is $V=\frac{1}{3}\pi {r}^{2}h$ for radius $r$ and height $h.$

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

If a polynomial of degree $n$ is divided by a binomial of degree 1, what is the degree of the quotient?

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$({x}^{2}+5x-1)\div (x-1)$

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(2{x}^{2}-9x-5)\div (x-5)$

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(3{x}^{2}+23x+14)\div (x+7)$

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(4{x}^{2}-10x+6)\div (4x+2)$

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(6{x}^{2}-25x-25)\div (6x+5)$

For the following exercises, use long division to divide. Specify the quotient and the remainder.

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

For the following exercises, use long division to divide. Specify the quotient and the remainder.

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

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$({x}^{3}-126)\div (x-5)$

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(3{x}^{2}-5x+4)\div (3x+1)$

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$({x}^{3}-3{x}^{2}+5x-6)\div (x-2)$

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(2{x}^{3}+3{x}^{2}-4x+15)\div (x+3)$

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.)

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