We continue to study polynomials. In this lesson we will learn how to divide them.

## Division of a polynomial by a monomial

To divide a polynomial by a monomial, divide each term of the polynomial by that monomial, then add the resulting quotients.

For example, divide the polynomial 15*x*^{2}*y*^{3 }+ 10*xy*^{2 }+ 5*xy*^{3} by the monomial xy. Write this division as a fraction:

Now divide each term of the polynomial 15*x*^{2}*y*^{3 }+ 10*xy*^{2 }+ 5*xy*^{3} by the monomial xy. The resulting quotients will be added together:

We obtained the usual division of monomials. Let's perform this division:

Thus, dividing the polynomial 15*x*^{2}*y*^{3 }+ 10*xy*^{2 }+ 5*xy*^{3} by the monomial xy results in the polynomial 15*xy*^{2 }+ 10*y* + 5*y*^{2}.

When dividing one number by another, the quotient must be such that when it is multiplied by the divisor, the result is the dividend. This rule also holds when dividing a polynomial by a monomial.

In our example, the product of the resulting polynomial 15*xy*^{2 }+ 10*y *+ 5*y*^{2} and the divisor xy should be equal to the polynomial 15*x*^{2}*y*^{3 }+ 10*xy*^{2 }+ 5*xy*^{3}, i.e. the original dividend. Let's check if this is true:

(15*xy*^{2 }+ 10*y *+ 5*y*^{2})*xy* = 15*x*^{2}*y*^{3 }+ 10*xy*^{2 }+ 5*xy*^{3}

Dividing a polynomial by a monomial is very similar to adding fractions with common denominators. We remember that to add fractions with identical denominators, you add their numerators and leave the denominator unchanged.

For example, to add fractions , , and , write the following expression:

If we calculate the expression , we get a fraction , the value of which is 1.5.

In this case, we can return expression to the initial state , and calculate each fraction separately, then add up the resulting quotients. The result will still be 1.5

The same thing happens when you divide a polynomial by a monomial. A monomial is the common denominator of all the terms of a polynomial. For example, dividing a polynomial **ax + bx + cx** by a polynomial x results in three fractions with a common denominator **x**

Calculating each fraction will result in a polynomial a + b + c

**Example 2.** Divide polynomial 8*m*^{3}*n *+ 24*m*^{2}*n*^{2} by monomial 8*m*^{2}*n*

**Example 3.** Divide polynomial 4*c*^{2}*d *β 12*c*^{4}*d*^{3} by monomial β4*c*^{2}*d*

## Division of a monomial by a polynomial

There is no identity transformation to divide a monomial by a polynomial.

Suppose we want to divide a monomial 2xy by a polynomial 5x + 3y + 5.

The result of this division should be a polynomial whose multiplication with the polynomial 5x + 3y + 5 gives a monomial 2xy. But there is no polynomial whose multiplication with the polynomial 5x + 3y + 5 would result in a monomial 2xy, since the multiplication of polynomials results in a polynomial, not a monomial.

But in the books you can find tasks to find the value of an expression with given values of the variables. In the initial expressions of such tasks, there may be a division of a monomial by a polynomial. In this case, no transformations need to be performed. It is enough to substitute the values of the variables in the original expression and calculate the resulting numerical expression.

For example, find the value of expression when x* *= 2.

Expression is a division of a monomial by a polynomial. In this case we will not be able to perform any transformations. The only thing we can do is substitute number 2 in the original expression instead of the x variable and find the value of the expression:

## Division of a polynomial by a polynomial

If you multiply the first polynomial by the second polynomial, you get a third polynomial. For example, if you multiply polynomial x + 5 by polynomial x + 3, you get polynomial x^{2} + 8x + 15

(*x* + 5)(*x* + 3) = *x*^{2} + 5*x* + 3*x* + 15 = *x*^{2} + 8*x* + 15

(*x* + 5)(*x* + 3) = *x*^{2} + 8*x* + 15

If you divide the product by the multiplier, you get the multiplicand. This rule applies not only to numbers, but also to polynomials.

Then according to this rule, dividing the polynomial x^{2} + 8x + 15 by the polynomial x + 3 should result in the polynomial x + 5.

Dividing a polynomial by a polynomial is done by of long division. The difference will be that when dividing a polynomial, you don't need to determine the first incomplete divisor, as in the case of dividing regular numbers.

Perform the long division of the polynomial x^{2} + 8x + 15 by the polynomial x + 3. In this way we will see step by step how the polynomial x + 5 is obtained.

In this case we know the result beforehand. It will be the polynomial x + 5. But more often than not, the result is unknown. So we will comment on the solution as if the result were unknown.

The result of division should be a new polynomial. Members of this polynomial will appear one by one in the process of division.

Now our task is to find the first term of the new polynomial. How do we do this?

When we initially multiplied the polynomials x + 5 and x + 3, we first multiplied the first term of the first polynomial by the first term of the second polynomial. In this way we got the first term of the third polynomial:

If we divide the first term of the third polynomial by the first term of the second polynomial, we get the first term of the first polynomial. This is what we need. After all, we have to arrive at the polynomial x + 5.

The same principle of finding the first term will be followed when solving other tasks on dividing polynomials.

So, to find the first term of a new polynomial, divide the first term of the dividend by the first term of the divisor.

If the first term of the dividend (in our case it is x^{2}) is divided by the first term of the divisor (which is x), we get x. That is, the first term of the new polynomial is x. Write it down at the right angle:

Now, as with dividing regular numbers, multiply x by the divisor x + 3. At this point, you should be able to multiply a monomial by a polynomial. When x is multiplied by x + 3, you get x^{2} + 3x. Write this polynomial under the divisor x^{2} + 8x + 15 so that the like terms are under each other:

Now subtract x^{2} + 8x + 15 from the dividend x^{2} + 3x. Subtract like terms from like terms. If you subtract x^{2} from x^{2}, you get 0. Zero is not written down. Next, if you subtract 3x from 8x, you get 5x. Write 5x so that the term is below the terms 3x and 8x

Now, as with dividing regular numbers, we take down the next term of the divisor. The next term is 15. It should be written down together with its sign:

Now divide the polynomial 5x + 15 by x + 3. To do this we need to find the second term of the new polynomial. To find it, divide the first term of the dividend (now it is the term 5x) by the first term of the divisor (that is the term x). If you divide 5x by x, you get 5. So the second term of the new polynomial is 5. Write it under the right angle, together with its sign (term 5 is positive in this case)

Now multiply 5 by the divisor x + 3. When you multiply 5 by x + 3, you get 5x + 15. Write this polynomial under the dividend 5x + 15

Now subtract 5x + 15 from the dividend 5x + 15. If you subtract 5x + 15 from 5x + 15 you get 0.

This completes division.

After division, you can check by multiplying the quotient by the divisor. In this case, if the quotient of x + 5 is multiplied by the divisor of x + 3, you should get the polynomial x^{2} + 8x + 15

(*x* + 5)(*x* + 3) = *x*^{2} + 5*x* + 3*x* + 15 = *x*^{2} + 8*x* + 15

**Example 2.** Divide the polynomial x^{2} - 8x + 7 by the polynomial x - 7

Write long division:

Find the first term of the quotient. Divide the first term of the dividend by the first term of the divisor, we get x. Write x at the right corner:

Multiply x by x - 7 to get x^{2} - 7x. Write this polynomial under the dividend x^{2} - 8x + 7, so that the like terms are located under each other:

Subtract x^{2} - 8x + 7 from the polynomial x^{2} - 7x. Subtracting x^{2} from x^{2} yields 0. Zero is not written down. And subtracting -7x from -8x gives -x, because -8x - (-7x) = -8x + 7x = -x. Write -x under the terms -7x and -8x. Then write down the next term 7

Be careful when subtracting negative terms. Often mistakes are made at this stage. If columnar subtraction is difficult at first, you can use the regular columnar subtraction of polynomials that we studied earlier. To do this, you need to write out the dividend and subtract from it the polynomial below it. The advantage of this method is that the next terms of the dividend don't need to be subtracted - they will automatically go into the new dividend. Let's use this method:

Let's return to our task. Let's divide the polynomial -x + 7 by x - 7. To do this we need to find the second term of the quotient. To find it, divide the first term of the dividend (now it is the term -x) by the first term of the divisor (that is the term x). If -x is divided by x, you get -1. Write -1 under the right-hand corner along with its sign:

Multiply -1 by x - 7 to get -x + 7. Write this polynomial under the dividend -x + 7

Now subtract -x + 7 from -x + 7. If you subtract -x + 7 from -x + 7 you get 0

The division is complete. Thus, the quotient of the division of the polynomial x^{2} - 8x + 7 by the polynomial x - 7 is x - 1

Let's check. Multiply the quotient x - 1 by the divisor x - 7. We should have a polynomial x^{2} - 8x + 7

(*x* β 1)(*x* β 7) = *x*^{2} β *x* β 7*x* + 7 = *x*^{2} β 8*x* + 7

**Example 3.** Divide the polynomial *x*^{6 }+ 2*x*^{4 }+ *x*^{7 }+ 2*x*^{5} by the polynomial *x*^{2 }+ *x*^{3}

Find the first term of the quotient. Divide the first term of the dividend by the first term of the divisor, we get x^{4}

Multiply x^{4} by the divisor *x*^{2 }+ *x*^{3} and write the result under the dividend. If x^{4} is multiplied by *x*^{2 }+ *x*^{3} you get *x*^{6 }+ *x*^{7}. Write the terms of this polynomial under the dividend, so that like terms are located under each other:

Now subtract the polynomial *x*^{6 }+ *x*^{7} from the dividend. Subtracting *x*^{6} from *x*^{6} will result in 0. Subtracting x^{7} from x^{7} will also result in 0. The remaining terms 2x^{4} and 2x^{5} will be written off:

A new dividend 2*x*^{4 }+ 2*x*^{5} is obtained. The same dividend could be obtained by writing out separately the polynomial *x*^{6 }+ 2*x*^{4 }+ *x*^{7 }+ 2*x*^{5} and subtracting from it the polynomial *x*^{6 }+ *x*^{7}

Divide the polynomial 2*x*^{4 }+ 2*x*^{5} by the divisor *x*^{2 }+ *x*^{3}. As before, first divide the first term of the dividend by the first term of the divisor, we get 2x^{2}. Write this term in the quotient:

Multiply 2x^{2} by the divisor *x*^{2 }+ *x*^{3} and write the result under the dividend. If 2x^{2} is multiplied by *x*^{2 }+ *x*^{3} you get 2*x*^{4 }+ 2*x*^{5}. Write the terms of this polynomial under the dividend, so that like terms are located under each other. Then do the subtraction:

Subtracting polynomial 2*x*^{4 }+ 2*x*^{5} from polynomial 2*x*^{4 }+ 2*x*^{5} resulted in 0, so the division was successful.

In the intermediate calculations the terms of the new dividend were located away from each other, forming large distances. This was because when multiplying the quotient by the divisor, the results were written so that like terms were located under each other.

These distances between terms of a new dividend are formed when the terms of the original polynomials are arranged disorderly. Therefore before division, it is desirable to order the terms of the original polynomials in descending order of powers. Then the solution will have a neater and clearer form.

Solve the previous example by arranging the terms of the original polynomials in descending order of powers. If we order the terms of the polynomial *x*^{6 }+ 2*x*^{4 }+ *x*^{7 }+ 2*x*^{5} in descending order of powers, we get the polynomial *x*^{7 }+ *x*^{6 }+ 2*x*^{5 }+ 2*x*^{4}. And if we order the terms of the polynomial *x*^{2 }+ *x*^{3} in descending order of powers, we get the polynomial *x*^{3 }+ *x*^{2}

Then the division of the polynomial *x*^{6 }+ 2*x*^{4 }+ *x*^{7 }+ 2*x*^{5} by the polynomial *x*^{2 }+ *x*^{3} will take the following form:

The division is complete. Thus, the quotient of the division of the polynomial *x*^{6 }+ 2*x*^{4 }+ *x*^{7 }+ 2*x*^{5} by the polynomial *x*^{2 }+ *x*^{3} is *x*^{4 }+ 2*x*^{2}

Let's check. Multiply the quotient *x*^{4 }+ 2*x*^{2} by the divisor *x*^{2 }+ *x*^{3}. We should have a polynomial *x*^{6 }+ 2*x*^{4 }+ *x*^{7 }+ 2*x*^{5}

(*x*^{4 }+ 2*x*^{2})(*x*^{2 }+ *x*^{3}) = *x*^{4 }(*x*^{2 }+ *x*^{3}) + 2*x*^{2}(*x*^{2 }+ *x*^{3}) = *x*^{6 }+ 2*x*^{4 }+ *x*^{7 }+ 2*x*^{5}

When multiplying polynomials, the terms of the original polynomials should also be ordered in descending order of powers. Then the terms of the resulting polynomial will also be ordered in descending order of powers.

Rewrite the multiplication of (*x*^{4 }+ 2*x*^{2})(*x*^{2 }+ *x*^{3}) by ordering the terms of the polynomials in descending order of powers.

(*x*^{4 }+ 2*x*^{2})(*x*^{3 }+ *x*^{2}) = *x*^{4}(*x*^{3 }+ *x*^{2}) + 2*x*^{2}(*x*^{3 }+ *x*^{2}) = *x*^{7 }+ *x*^{6 }+ 2*x*^{5 }+ 2*x*^{4}

**Example 4.** Divide the polynomial 17*x*^{2 }β 6*x*^{4 }+ 5*x*^{3 }β 23*x* + 7 by the polynomial 7 β 3*x*^{2 }β 2*x*

Arrange the terms of the original polynomials in descending order of powers and perform this division:

So,

**Example 5.** Divide polynomial 4*a*^{4 }β 14*a*^{3}*b* β 24*a*^{2}*b*^{2 }β 54*b*^{4} by polynomial *a*^{2 }β 3*ab* β 9*b*^{2}

Find the first term of the quotient. Divide the first term of the dividend by the first term of the divisor, we get 4a^{2}. Write 4a^{2} in the quotient:

Multiply 4a^{2} by the divisor a^{2} - 3ab - 9b^{2} and write the result under the dividend:

Subtract the resulting polynomial 4*a*^{4 }β 12*a*^{3}*b *β 36*a*^{2}*b*^{2} from the dividend.

Now divide β2*a*^{3}*b *+ 12*a*^{2}*b*^{2 }β 54*b*^{4} by the divisor *a*^{2 }β 3*ab* β 9*b*^{2}. Divide the first term of the dividend by the first term of the divisor, we get -2ab. Write -2ab in the quotient:

Multiply -2ab by the divisor *a*^{2 }β 3*ab* β 9*b*^{2} and write the result under the dividend β2*a*^{3}*b *+ 12*a*^{2}*b*^{2 }β 54*b*^{4}

Subtract β2*a*^{3}*b *+ 12*a*^{2}*b*^{2 }β 54*b*^{4} from the polynomial β2*a*^{3}*b *+ 12*a*^{2}*b*^{2 }β 18*ab*^{3}. When subtracting like terms, we find that -54b^{4} and 18ab^{3} are not like terms, so their subtraction will not give any transformation. In this case we deduct where it is possible, namely, we subtract β2*a*^{3}*b* from β2*a*^{3}*b* and 6*a*^{2}*b*^{2} from 12*a*^{2}*b*^{2}, and write down the subtraction of 18ab^{3} from -54b^{4} as the difference β54*b*^{4 }β (+18*ab*^{3}) or β54*b*^{4 }β 18*ab*^{3}

The same result can be obtained if you perform the subtraction of polynomials to a string using brackets:

Let's return to our task. Divide 6*a*^{2}*b*^{2 }β 54*b*^{4 }β 18*ab*^{3} by the divisor *a*^{2 }β 3*ab* β 9*b*^{2}. Divide the first term of the dividend by the first term of the divisor, we get 6b^{2}. Write 6b^{2} in the quotient:

Multiply 6b^{2} by the divisor *a*^{2 }β 3*ab* β 9*b*^{2} and write the result under the dividend 6*a*^{2}*b*^{2 }β 54*b*^{4 }β 18*ab*^{3}. Immediately subtract this result from the dividend 6*a*^{2}*b*^{2 }β 54*b*^{4 }β 18*ab*^{3}

The division is complete. Thus, the quotient of the division of the polynomial 4*a*^{4 }β 14*a*^{3}*b* β 24*a*^{2}*b*^{2 }β 54*b*^{4} by the polynomial *a*^{2 }β 3*ab* β 9*b*^{2} is 4*a*^{2 }β 2*ab* + 6*b*^{2}.

Let's check. Multiply the quotient 4*a*^{2 }β 2*ab* + 6*b*^{2} by the divisor *a*^{2 }β 3*ab* β 9*b*^{2}. We should have a polynomial 4*a*^{4 }β 14*a*^{3}*b* β 24*a*^{2}*b*^{2 }β 54*b*^{4}

## Division of a polynomial by a polynomial with remainder

As with dividing regular numbers, dividing a polynomial by a polynomial can produce a remainder.

First, let's recall the division of regular numbers with a remainder. For example, let's divide 15 by 2. With a remainder this division will be done as follows:

That is, when you divide 15 by 2 you get 7 integers and 1 in the remainder. The answer is written as follows:

A rational number is read as seven whole numbers plus one second. The plus sign is traditionally not written down. But if dividing a polynomial by a polynomial produces a remainder, then this plus must be written.

For example, if you divide a polynomial **a** by a polynomial **b** to get a quotient **c** and leave a remainder **q**, then the answer will be written as follows:

For example, divide the polynomial 2*x*^{3 }β *x*^{2 }β 5*x *+ 4 by the polynomial x - 3

Find the first term of the quotient. Divide the first term of the dividend by the first term of the divisor, we get 2x^{2}. Write 2x^{2} in the quotient:

Multiply 2x^{2} by the divisor x - 3 and write the result under the dividend:

Subtract the resulting polynomial 2*x*^{3 }β 6*x*^{2} from the dividend.

Now divide 5x^{2} - 5x + 4 by the divisor x - 3. Divide the first term of the dividend by the first term of the divisor, we get 5x. Write 5x in the quotient:

Multiply 5x by the divisor x - 3 and write the result under the dividend 5*x*^{2 }β 5*x *+ 4

Subtract from the polynomial 5*x*^{2 }β 5*x *+ 4 the polynomial 5x^{2} - 15x

Now divide 10x + 4 by the divisor x - 3. Divide the first term of the dividend by the first term of the divisor, we get 10. Write 10 in the quotient:

Multiply 10 by the divisor x - 3 and write the result under the dividend 10x + 4. Immediately subtract this result from the dividend 10x + 4

The number 34 resulting from subtracting the polynomial 10x - 30 from the polynomial 10x + 4 is the remainder. We cannot find the next term of the quotient which, when multiplied with a divisor of x - 3, would give us 34 as a result.

Therefore, dividing the polynomial 2*x*^{3 }β 2*x*^{2 }β 5*x *+ 4 by the polynomial x - 3 yields 2*x*^{2 }+ 5*x *+ 10 and 34 in the remainder. The answer is written in the same way as when dividing regular numbers. First write down the integer part (it is located under the right corner) plus the remainder divided by the divisor:

## When division of polynomials is impossible

Dividing a polynomial by a polynomial is impossible if the power of the dividend is less than the power of the divisor.

For example, you cannot divide a polynomial x^{3} + x by a polynomial *x*^{4} + *x*^{2} because the dividend is a third power polynomial and the divisor is a fourth power polynomial.

Contrary to this prohibition, you can try to divide the polynomial x^{3} + x by the polynomial *x*^{4} + *x*^{2}, and even get the quotient x^{β}^{1}, which when multiplied with the divisor will give the dividend:

But dividing a polynomial by a polynomial should result in a polynomial, and the quotient x^{β}^{1} is not a polynomial. After all, a polynomial consists of a monomial, and a monomial is the **product** of numbers, variables, and powers. The expression x^{β}^{1} is a fraction ,which is not a product.

Suppose there is a rectangle with sides 4 and 2

The area of this rectangle will be 4 Γ 2 = 8 square units.

Let's increase the length and width of this rectangle by **x**

Let's complete the missing sides:

Now the rectangle has length x + 4 and width x + 2. The area of this rectangle is equal to the product (x + 4)(x + 2) and is expressed by the polynomial *x*^{2 }+ 6*x *+ 8

(*x *+ 4)(*x *+ 2) = *x*^{2 }+ 4*x *+ 2*x *+ 8 = *x*^{2 }+ 6*x *+ 8

In doing so, we can perform the inverse operation, namely, divide the area x^{2} + 6x + 8 by the width x + 2 and obtain the length x + 4.

The power of a polynomial x^{2} + 6x + 8 is equal to the sum of powers of polynomials x + 4 and x + 2, so none of the powers of a polynomial can exceed the power of the product polynomial. Consequently, for inverse division to be possible, the power of the divisor must be less than the power of the dividend.

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