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Dividing Polynomials

Use long division to divide polynomials.

Using Long Division to Divide Polynomials

We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let’s divide 178 by 3 using long division.

Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.

\[\begin{array}{l}\text{dividend }=(\text{divisor }⋅\text{ quotient})+\text{ remainder} \\ 178=(3⋅59)+1 \\ =177+1 \\ =178\end{array}\]

We call this the Division Algorithm and will discuss it more formally after looking at an example.

Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide \(2{x}^{3}-3{x}^{2}+4x+5\) by \(x+2\) using the long division algorithm, it would look like this:

We have found

\[\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\frac{31}{x+2}\]

or

\[2{x}^{3}-3{x}^{2}+4x+5=(x+2)(2{x}^{2}-7x+18)-31\]

We can identify the dividend, the divisor, the quotient, and the remainder.

Condensed: the full section is in OpenStax Precalculus 2e.

Using Synthetic Division to Divide Polynomials

As we’ve seen, long division of polynomials can involve many steps and be quite cumbersome. Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.

To illustrate the process, recall the example at the beginning of the section.

Divide \(2{x}^{3}-3{x}^{2}+4x+5\) by \(x+2\) using the long division algorithm.

The final form of the process looked like this:

There is a lot of repetition in the table. If we don’t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.

Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the “divisor” to –2, multiply and add. The process starts by bringing down the leading coefficient.

Example

Try it.

Use synthetic division to divide \(5{x}^{2}-3x-36\) by \(x-3.\)

Solution

Begin by setting up the synthetic division. Write \(k\) and the coefficients.

Bring down the lead coefficient. Multiply the lead coefficient by \(k.\)

Continue by adding the numbers in the second column. Multiply the resulting number by \(k.\) Write the result in the next column. Then add the numbers in the third column.

The result is \(5x+12.\) The remainder is 0. So \(x-3\) is a factor of the original polynomial.

Example

Try it.

Use synthetic division to divide \(-9{x}^{4}+10{x}^{3}+7{x}^{2}-6\) by \(x-1.\)

Solution

Notice there is no x-term. We will use a zero as the coefficient for that term.

The result is \(-9{x}^{3}+{x}^{2}+8x+8+\frac{2}{x-1}.\)

Condensed: the full section is in OpenStax Precalculus 2e.

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.

Example

Try it.

The volume of a rectangular solid is given by the polynomial \(3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.\) The length of the solid is given by \(3x\) and the width is given by \(x-2.\) Find the height of the solid.

Solution

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 .

We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.

\[\begin{array}{l}\ V=l⋅w⋅h \\ 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x⋅(x-2)⋅h\end{array}\]

To solve for \(h,\) first divide both sides by \(3x.\)

\[\begin{array}{l}\frac{3x⋅(x-2)⋅h}{3x}=\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x} \\ (x-2)h={x}^{3}-{x}^{2}-11x+18\end{array}\]

Now solve for \(h\) using synthetic division.

\[h=\frac{{x}^{3}-{x}^{2}-11x+18}{x-2}\]\[\begin{array}{lllllll} \\ 2\begin{array}{llll}1 & -1 & -11 & 18 \\ & 2 & 2 & -18\end{array} \\ \begin{array}{llll}\ 1 & \ 1 & \ -9 & \ 0\end{array}\end{array}\]

The quotient is \({x}^{2}+x-9\) and the remainder is 0. The height of the solid is \({x}^{2}+x-9.\)

Key Concepts

  • Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See and .
  • 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 \(x-k.\) See , , and .
  • Polynomial division can be used to solve application problems, including area and volume. See .

Dividing Polynomials

  • Dividing polynomials using long division (IA 5.4.3)
  • Dividing polynomials using synthetic division (IA 5.4.4)

To divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So, let’s look carefully at the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

Example

Try it.

When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder.

Vocabulary of the example.

Fill in the blanks.

Try it.

When dividing 69 by 4,

the dividend is_________,

the divisor is________,

the quotient is________,

and the remainder is________.

We check division by multiplying the quotient by the divisor and adding the remainder.

Example

Try it.

Find the quotient: \(({x}^{2}+9x+20)\div (x+5).\)

Solution
Write it as a long division problem.
Be sure the dividend is written in descending order of powers, with no missing terms.
Divide \({x}^{2}\) by \(x.\) It may help to ask yourself, “What do I need
to multiply \(x\) by to get \({x}^{2}\) ?”
Put the answer, \(x,\) in the quotient over the \(x\) term.
Multiply \(x\) times \(x+5.\) Line up the like terms under the dividend.
Subtract \({x}^{2}+5x\) from \({x}^{2}+9x.\)
You may find it easier to change the signs and then add.
Then bring down the last term, 20.

Divide \(4x\) by \(x.\) It may help to ask yourself, “What do I
need to multiply \(x\) by to get \(4x\) ?”
Put the answer, \(4\) , in the quotient over the constant term.
Multiply 4 times \(x+5.\)
Subtract \(4x+20\) from \(4x+20.\)
Check:
Multiply the quotient by the divisor. \(\ (x+4)(x+5)\)
You should get the dividend. \(\ {x}^{2}+9x+20✓\)

Try it.

Divide using long division of polynomials: \(({x}^{2}+10x+21)\div (x+3)\)

Sometimes division of polynomials, just like division of numbers, leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

Also, if you look back at the dividends in previous examples, you will notice that the terms were written in descending order of degrees, and there were no missing degrees.

Example

Try it.

\((5x+{x}^{4}-{x}^{2}-6)\div (x+2)\)

Solution

Notice, this polynomial is not in descending order and it is missing \({x}^{3}\) term. We need to write it in the correct order and add \(0{x}^{3}\) as a placeholder.

To check, multiply divisor by the quotient and add remainder \((x+2)({x}^{3}-2{x}^{2}+3x-1)-4\)

The result should be \({x}^{4}-{x}^{2}+5x-6\)

Dividing polynomials using long division.

Try it.

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

Try it.

\(({x}^{3}-8)\div (x-2)\)

Condensed: the full section is in OpenStax College Algebra 2e.

Using Long Division to Divide Polynomials

We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let’s divide 178 by 3 using long division.

Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.

\[\begin{array}{lll}\text{dividend} & = & (\text{divisor}⋅\text{quotient) }+\text{ remainder} \\ 178 & = & (3⋅59)+1 \\ & = & 177+1 \\ & = & 178\end{array}\]

We call this the Division Algorithm and will discuss it more formally after looking at an example.

Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide \(2{x}^{3}-3{x}^{2}+4x+5\) by \(x+2\) using the long division algorithm, it would look like this:

We have found

\[\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\frac{31}{x+2}\]

or

\[2{x}^{3}-3{x}^{2}+4x+5=(x+2)(2{x}^{2}-7x+18)-31\]

We can identify the dividend, the divisor, the quotient, and the remainder.

Condensed: the full section is in OpenStax College Algebra 2e.

Using Synthetic Division to Divide Polynomials

As we’ve seen, long division of polynomials can involve many steps and be quite cumbersome. Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.

To illustrate the process, recall the example at the beginning of the section.

Divide \(2{x}^{3}-3{x}^{2}+4x+5\) by \(x+2\) using the long division algorithm.

The final form of the process looked like this:

There is a lot of repetition in the table. If we don’t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.

Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the “divisor” to –2, multiply and add. The process starts by bringing down the leading coefficient.

We then multiply it by the “divisor” and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is \(2{x}^{2}-7x+18\) and the remainder is \(-31.\) The process will be made more clear in .

Example

Try it.

Use synthetic division to divide \(4{x}^{3}+10{x}^{2}-6x-20\) by \(x+2.\)

Solution

The binomial divisor is \(x+2\) so \(k=-2.\) Add each column, multiply the result by –2, and repeat until the last column is reached.

The result is \(4{x}^{2}+2x-10.\) The remainder is 0. Thus, \(x+2\) is a factor of \(4{x}^{3}+10{x}^{2}-6x-20.\)

Condensed: the full section is in OpenStax College Algebra 2e.

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.

Example

Try it.

The volume of a rectangular solid is given by the polynomial \(3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.\) The length of the solid is given by \(3x\) and the width is given by \(x-2.\) Find the height, \(h,\) of the solid.

Solution

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 .

We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.

\[\begin{array}{lll}V & = & l⋅w⋅h \\ 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x & = & 3x⋅(x-2)⋅h\end{array}\]

To solve for \(h,\) first divide both sides by \(3x.\)

\[\begin{array}{lll}\frac{3x⋅(x-2)⋅h}{3x} & = & \frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x} \\ (x-2)h & = & {x}^{3}-{x}^{2}-11x+18\end{array}\]

Now solve for \(h\) using synthetic division.

\[h=\frac{{x}^{3}-{x}^{2}-11x+18}{x-2}\]

The quotient is \({x}^{2}+x-9\) and the remainder is 0. The height of the solid is \({x}^{2}+x-9.\)

Key Concepts

  • Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See and .
  • 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 \(x-k.\) See , , and .
  • Polynomial division can be used to solve application problems, including area and volume. See .

अब तपाईँ कुनै गणकयन्त्रले यो एकलाई समाधान गर्दैन, तर यसको टुक्राहरू गणनायोग्य छन् । तल एउटा प्रयास गर्नुहोस्, वा तपाईँको आफ्नै टाइप गर्नुहोस् ।

अभ्यास (40)

कागजमा पहिला प्रत्येक प्रयास गर्नुहोस् । जाँच गर्न जवाफ प्रकट गर्नुहोस्; रुजु गरिएकाहरू प्रत्येक चरणका लागि समाधानकर्तामा खोल्न सकिन्छ ।

  1. Divide \(5{x}^{2}+3x-2\) by \(x+1.\)

    जवाफ प्रकट गर्नुहोस्

    The quotient is \(5x-2.\) The remainder is 0. We write the result as

    \[\frac{5{x}^{2}+3x-2}{x+1}=5x-2\]

    or

    \[5{x}^{2}+3x-2=(x+1)(5x-2)\]

  2. Divide \(6{x}^{3}+11{x}^{2}-31x+15\) by \(3x-2.\)

    जवाफ प्रकट गर्नुहोस्

    There is a remainder of 1. We can express the result as:

    \[\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x-2}=2{x}^{2}+5x-7+\frac{1}{3x-2}\]

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

    जवाफ प्रकट गर्नुहोस्

    \(4{x}^{2}-8x+15-\frac{78}{4x+5}\)

  4. Use synthetic division to divide \(5{x}^{2}-3x-36\) by \(x-3.\)

    जवाफ प्रकट गर्नुहोस्

    Begin by setting up the synthetic division. Write \(k\) and the coefficients.

    Bring down the lead coefficient. Multiply the lead coefficient by \(k.\)

    Continue by adding the numbers in the second column. Multiply the resulting number by \(k.\) Write the result in the next column. Then add the numbers in the third column.

    The result is \(5x+12.\) The remainder is 0. So \(x-3\) is a factor of the original polynomial.

  5. Use synthetic division to divide \(4{x}^{3}+10{x}^{2}-6x-20\) by \(x+2.\)

    जवाफ प्रकट गर्नुहोस्

    The binomial divisor is \(x+2\) so \(k=-2.\) Add each column, multiply the result by –2, and repeat until the last column is reached.

    The result is \(4{x}^{2}+2x-10.\) The remainder is 0. Thus, \(x+2\) is a factor of \(4{x}^{3}+10{x}^{2}-6x-20.\)

  6. Use synthetic division to divide \(-9{x}^{4}+10{x}^{3}+7{x}^{2}-6\) by \(x-1.\)

    जवाफ प्रकट गर्नुहोस्

    Notice there is no x-term. We will use a zero as the coefficient for that term.

    The result is \(-9{x}^{3}+{x}^{2}+8x+8+\frac{2}{x-1}.\)

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

    जवाफ प्रकट गर्नुहोस्

    \(3{x}^{3}-3{x}^{2}+21x-150+\frac{1,090}{x+7}\)

  8. The volume of a rectangular solid is given by the polynomial \(3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.\) The length of the solid is given by \(3x\) and the width is given by \(x-2.\) Find the height of the solid.

    जवाफ प्रकट गर्नुहोस्

    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 .

    We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.

    \[\begin{array}{l}\ V=l⋅w⋅h \\ 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x⋅(x-2)⋅h\end{array}\]

    To solve for \(h,\) first divide both sides by \(3x.\)

    \[\begin{array}{l}\frac{3x⋅(x-2)⋅h}{3x}=\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x} \\ (x-2)h={x}^{3}-{x}^{2}-11x+18\end{array}\]

    Now solve for \(h\) using synthetic division.

    \[h=\frac{{x}^{3}-{x}^{2}-11x+18}{x-2}\]\[\begin{array}{lllllll} \\ 2\begin{array}{llll}1 & -1 & -11 & 18 \\ & 2 & 2 & -18\end{array} \\ \begin{array}{llll}\ 1 & \ 1 & \ -9 & \ 0\end{array}\end{array}\]

    The quotient is \({x}^{2}+x-9\) and the remainder is 0. The height of the solid is \({x}^{2}+x-9.\)

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

    जवाफ प्रकट गर्नुहोस्

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

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

    जवाफ प्रकट गर्नुहोस्

    The binomial is a factor of the polynomial.

  11. If a polynomial of degree \(n\) is divided by a binomial of degree 1, what is the degree of the quotient?

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

    जवाफ प्रकट गर्नुहोस्

    \(x+6+\frac{5}{x-1}\), quotient: \(x+6\), remainder: \(5\)

  13. \((2{x}^{2}-9x-5)\div (x-5)\)

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

    जवाफ प्रकट गर्नुहोस्

    \(3x+2\), quotient:  \(3x+2\), remainder:  \(0\)

  15. \((4{x}^{2}-10x+6)\div (4x+2)\)

  16. \((6{x}^{2}-25x-25)\div (6x+5)\)

    जवाफ प्रकट गर्नुहोस्

    \(x-5\), quotient: \(x-5\), remainder: \(0\)

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

  18. \((2{x}^{2}-3x+2)\div (x+2)\)

    जवाफ प्रकट गर्नुहोस्

    \(2x-7+\frac{16}{x+2}\), quotient: \(2x-7\), remainder: \(16\)

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

  20. \((3{x}^{2}-5x+4)\div (3x+1)\)

    जवाफ प्रकट गर्नुहोस्

    \(x-2+\frac{6}{3x+1}\), quotient: \(x-2\), remainder: \(6\)

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

  22. \((2{x}^{3}+3{x}^{2}-4x+15)\div (x+3)\)

    जवाफ प्रकट गर्नुहोस्

    \(2{x}^{2}-3x+5\), quotient: \(2{x}^{2}-3x+5\), remainder: \(0\)

  23. \((3{x}^{3}-2{x}^{2}+x-4)\div (x+3)\)

  24. \((2{x}^{3}-6{x}^{2}-7x+6)\div (x-4)\)

    जवाफ प्रकट गर्नुहोस्

    \(2{x}^{2}+2x+1+\frac{10}{x-4}\)

  25. \((6{x}^{3}-10{x}^{2}-7x-15)\div (x+1)\)

  26. \((4{x}^{3}-12{x}^{2}-5x-1)\div (2x+1)\)

    जवाफ प्रकट गर्नुहोस्

    \(2{x}^{2}-7x+1-\frac{2}{2x+1}\)

  27. \((9{x}^{3}-9{x}^{2}+18x+5)\div (3x-1)\)

  28. \((3{x}^{3}-2{x}^{2}+x-4)\div (x+3)\)

    जवाफ प्रकट गर्नुहोस्

    \(3{x}^{2}-11x+34-\frac{106}{x+3}\)

  29. \((-6{x}^{3}+{x}^{2}-4)\div (2x-3)\)

  30. \((2{x}^{3}+7{x}^{2}-13x-3)\div (2x-3)\)

    जवाफ प्रकट गर्नुहोस्

    \({x}^{2}+5x+1\)

  31. \((3{x}^{3}-5{x}^{2}+2x+3)\div (x+2)\)

  32. \((4{x}^{3}-5{x}^{2}+13)\div (x+4)\)

    जवाफ प्रकट गर्नुहोस्

    \(4{x}^{2}-21x+84-\frac{323}{x+4}\)

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

  34. \(({x}^{3}-21{x}^{2}+147x-343)\div (x-7)\)

    जवाफ प्रकट गर्नुहोस्

    \({x}^{2}-14x+49\)

  35. \(({x}^{3}-15{x}^{2}+75x-125)\div (x-5)\)

  36. \((9{x}^{3}-x+2)\div (3x-1)\)

    जवाफ प्रकट गर्नुहोस्

    \(3{x}^{2}+x+\frac{2}{3x-1}\)

  37. \((6{x}^{3}-{x}^{2}+5x+2)\div (3x+1)\)

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

    जवाफ प्रकट गर्नुहोस्

    \({x}^{3}-3x+1\)

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

  40. \(({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6)\div (x+3)\)

    जवाफ प्रकट गर्नुहोस्

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

तपाईँको आफ्नै काम गर्ने राख्नुहोस्

एक निःशुल्क खाता प्रत्येक पाठ मा नोट थप्छ, तपाईं के समाप्त गरेको छ को एक रेकर्ड, एक ठाउँमा आफ्नो समाधान समस्या, र एक शिक्षक तपाईं यो पृष्ठ बारेमा सोध्न सक्नुहुन्छ. गणित आफैलाई सबैलाई खुला छ, मा साइन इन वा छैन.

साइन अप गर्नुहोस् लगइन

यहाँ प्रयोग गरिएको प्रतीक

पूर्ण परिभाषा, एउटा तस्वीर, र यो प्रत्येक अक्षर अर्थ के लागि कुनै पनि प्रतीक ट्याप गर्नुहोस्।

कसरी: Dividing Polynomials

  1. Use long division to divide polynomials.
  2. Use synthetic division to divide polynomials.
  3. Set up the division problem.
  4. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
  5. Multiply the answer by the divisor and write it below the like terms of the dividend.
  6. Subtract the bottom
  7. Bring down the next term of the dividend.
  8. Repeat steps 2–5 until reaching the last term of the dividend.

प्रश्नहरू मानिसहरूले सोध्छन्

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

यो पृष्ठको भागहरू यसबाट अनुकूलित गरिएको छ OpenStax College Algebra 2e (CC BY-NC-SA 4.0), OpenStax Precalculus 2e (CC BY-NC-SA 4.0). यहाँको मुख्य पेशा कृषि र पशुपालन हो। यसको स्रोत गैर-व्यावसायिक प्रयोगका लागि लाइसेन्स गरिएको छ, यो पाठ निःशुल्क छ, केही पनि बिक्री गर्दैन, र उही लाइसेन्स अन्तर्गत बाँडिएको छ।

यसमा थप Precalculus

Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theoremParametric equations and polar coordinates