Using synthetic division to divide a second-degree polynomial
Use synthetic division to divide
$\text{\hspace{0.17em}}5{x}^{2}-3x-36\text{\hspace{0.17em}}$ by
$\text{\hspace{0.17em}}x-3.\text{\hspace{0.17em}}$
Begin by setting up the synthetic division. Write
$\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ and the coefficients.
Bring down the lead coefficient. Multiply the lead coefficient by
$\text{\hspace{0.17em}}k.\text{\hspace{0.17em}}$
Continue by adding the numbers in the second column. Multiply the resulting number by
$\text{\hspace{0.17em}}k.\text{\hspace{0.17em}}$ Write the result in the next column. Then add the numbers in the third column.
The result is
$\text{\hspace{0.17em}}5x+12.\text{\hspace{0.17em}}$ The remainder is 0. So
$\text{\hspace{0.17em}}x-3\text{\hspace{0.17em}}$ is a factor of the original polynomial.
Using synthetic division to divide a third-degree polynomial
Use synthetic division to divide
$\text{\hspace{0.17em}}4{x}^{3}+10{x}^{2}-6x-20\text{\hspace{0.17em}}$ by
$\text{\hspace{0.17em}}x+2.\text{\hspace{0.17em}}$
The binomial divisor is
$\text{\hspace{0.17em}}x+2\text{\hspace{0.17em}}$ so
$\text{\hspace{0.17em}}k=\mathrm{-2.}\text{\hspace{0.17em}}$ Add each column, multiply the result by –2, and repeat until the last column is reached.
The result is
$\text{\hspace{0.17em}}4{x}^{2}+2x-10.\text{\hspace{0.17em}}$ The remainder is 0. Thus,
$\text{\hspace{0.17em}}x+2\text{\hspace{0.17em}}$ is a factor of
$\text{\hspace{0.17em}}4{x}^{3}+10{x}^{2}-6x-20.\text{\hspace{0.17em}}$
Using synthetic division to divide a fourth-degree polynomial
Use synthetic division to divide
$\text{\hspace{0.17em}}-9{x}^{4}+10{x}^{3}+7{x}^{2}-6\text{\hspace{0.17em}}$ by
$\text{\hspace{0.17em}}x-1.\text{\hspace{0.17em}}$
Notice there is no
x -term. We will use a zero as the coefficient for that term.
The result is
$\text{\hspace{0.17em}}-9{x}^{3}+{x}^{2}+8x+8+\frac{2}{x-1}.$
Using polynomial division to solve application problems
Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.
Using polynomial division in an application problem
The volume of a rectangular solid is given by the polynomial
$\text{\hspace{0.17em}}3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.\text{\hspace{0.17em}}$ The length of the solid is given by
$\text{\hspace{0.17em}}3x\text{\hspace{0.17em}}$ and the width is given by
$\text{\hspace{0.17em}}x-2.\text{\hspace{0.17em}}$ Find the height,
$\text{\hspace{0.17em}}t,$ of the solid.
There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch as in
[link] .
We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.
Now solve for
$\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ using synthetic division.
$$h=\frac{{x}^{3}-{x}^{2}-11x+18}{x-2}$$
The quotient is
$\text{\hspace{0.17em}}{x}^{2}+x-9\text{\hspace{0.17em}}$ and the remainder is 0. The height of the solid is
$\text{\hspace{0.17em}}{x}^{2}+x-9.$
The area of a rectangle is given by
$\text{\hspace{0.17em}}3{x}^{3}+14{x}^{2}-23x+6.\text{\hspace{0.17em}}$ The width of the rectangle is given by
$\text{\hspace{0.17em}}x+6.\text{\hspace{0.17em}}$ Find an expression for the length of the rectangle.
Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See
[link] and
[link].
The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form
$\text{\hspace{0.17em}}x-k.\text{\hspace{0.17em}}$ See
[link],[link], and
[link].
Polynomial division can be used to solve application problems, including area and volume. See
[link].
Questions & Answers
show that the set of all natural number form semi group under the composition of addition
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the
fraction, the value of the fraction becomes 2/3. Find the original fraction.
2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
Q2
x+(x+2)+(x+4)=60
3x+6=60
3x+6-6=60-6
3x=54
3x/3=54/3
x=18
:. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Solve for the first variable in one of the equations, then substitute the result into the other equation.
Point For:
(6111,4111,−411)(6111,4111,-411)
Equation Form:
x=6111,y=4111,z=−411x=6111,y=4111,z=-411