5.4 Dividing Polynomials

Divide $16 x^{3} - 12 x^{2} + 20 x - 3$ by $4 x + 5.$

Use synthetic division to divide $3 x^{4} + 18 x^{3} - 3 x + 40$ by $x + 7.$

The area of a rectangle is given by $3 x^{3} + 14 x^{2} - 23 x + 6.$ The width of the rectangle is given by $x + 6.$ Find an expression for the length of the rectangle.

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

Use long division to divide. Specify the quotient and the remainder.

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

Use long division to divide. Specify the quotient and the remainder.

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

Use long division to divide. Specify the quotient and the remainder.

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

Use long division to divide. Specify the quotient and the remainder.

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

Use long division to divide. Specify the quotient and the remainder.

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

Use long division to divide. Specify the quotient and the remainder.

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

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.)

$\left(2 x^{3} - 6 x^{2} - 7 x + 6\right) \div \left(x - 4\right)$

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.)

$\left(4 x^{3} - 12 x^{2} - 5 x - 1\right) \div \left(2 x + 1\right)$

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.)

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

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.)

$\left(2 x^{3} + 7 x^{2} - 13 x - 3\right) \div \left(2 x - 3\right)$

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.)

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

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.)

$\left(x^{3} - 21 x^{2} + 147 x - 343\right) \div \left(x - 7\right)$

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.)

$\left(9 x^{3} - x + 2\right) \div \left(3 x - 1\right)$

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.)

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

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.)

$\left(x^{4} + 2 x^{3} - 3 x^{2} + 2 x + 6\right) \div \left(x + 3\right)$

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.)

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

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.)

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

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.)

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

Use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

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

Use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

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

Use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

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

Use synthetic division to find the quotient and remainder.

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

Use synthetic division to find the quotient and remainder.

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

Use synthetic division to find the quotient and remainder.

$\frac{x^{4} - 22}{x + 2}$

Use a calculator with CAS to answer the questions.

Consider $\frac{x^{k} + 1}{x + 1}$ for $k = 1 , 3 , 5.$ What do you expect the result to be if $k = 7 ?$

Use a calculator with CAS to answer the questions.

Consider $\frac{x^{k}}{x + 1}$ with $k = 1 , 2 , 3.$ What do you expect the result to be if $k = 4 ?$

Use synthetic division to determine the quotient involving a complex number.

$\frac{x + 1}{x - i}$

Use synthetic division to determine the quotient involving a complex number.

$\frac{x + 1}{x + i}$

Use synthetic division to determine the quotient involving a complex number.

$\frac{x^{3} + 1}{x - i}$

Use the given length and area of a rectangle to express the width algebraically.

Length is $2 x + 5 ,$ area is $4 x^{3} + 10 x^{2} + 6 x + 15$

Use the given volume of a box and its length and width to express the height of the box algebraically.

Volume is $12 x^{3} + 20 x^{2} - 21 x - 36 ,$ length is $2 x + 3 ,$ width is $3 x - 4.$

Use the given volume of a box and its length and width to express the height of the box algebraically.

Volume is $10 x^{3} + 27 x^{2} + 2 x - 24 ,$ length is $5 x – 4 ,$ width is $2 x + 3.$

Use the given volume and radius of a cylinder to express the height of the cylinder algebraically.

Volume is $\pi \left(25 x^{3} - 65 x^{2} - 29 x - 3\right) ,$ radius is $5 x + 1.$

Use the given volume and radius of a cylinder to express the height of the cylinder algebraically.

Volume is $\pi \left(3 x^{4} + 24 x^{3} + 46 x^{2} - 16 x - 32\right) ,$ radius is $x + 4.$