Comparing Methods Long Division Again Answer

In this explainer, we will acquire how to detect the caliber and remainder when polynomials are divided, including the instance when the divisor is irreducible.

As with integers, dividing a polynomial (the dividend) past a divisor gives a caliber and a remainder .

Think that a polynomial is a finite sum of monomials which has nonnegative exponents. Hence, expressions of the forms , , and eight are all examples of polynomials, whereas expressions such every bit , , and are not polynomial expressions. In this explainer, we volition focus on dividing polynomials of one variable.

Usually when considering the division of polynomials, we write rather than . We can call back of long division equally finding polynomials and such that and nosotros say that that the division yields a quotient and a remainder .

We tin write this equivalently as a multiplication equation as follows:

However, not all equations in this form are division equations. For instance, consider the equation

This can be written as only it does not authorize as division past considering, just as with integer division, the remainder must always have a lower caste than the divisor.

A right division equation, in this example, would be

The remainder is 35 which has caste 0, which is less than the degree of which is ane.

When we use the division algorithm to get an of degree less than , the quotient and the remainder are uniquely determined. We will now outline the partitioning algorithm we can use to find and .

Long sectionalization of polynomials is much the same equally long division for integers: at each step, we compare the leading coefficient of the divisor with the electric current residue, which starts off being the dividend itself. The objective at each step is to remove this leading term. Let us look at an example of how to practice this.

We will employ the example of dividing by to demonstrate the method.

In the commencement stride, we split the term of the highest degree in the dividend past the term of the highest degree in the divisor. Hence, we divide by to get .

We write the consequence of this partitioning above the line.

Nosotros at present multiply this term past the divisor and write the result below the dividend so that the terms of equal caste marshal.

We now subtract the resulting expression from the dividend.

This should consequence in the states eliminating the term with the highest degree. We tin then bring downwards the terms from the dividend to get an expression for our first residual. If this is of equal or college degree than the divisor, as is the case here, we echo this process again.

Hence, we carve up the terms of highest degree. That is we divide by to get 13.

We write this higher up the line next to our terminal term.

Nosotros at present multiply this term past the divisor and write the effect beneath the dividend so that the terms of equal caste align.

We now subtract the resulting expression from the first residue.

This should event in us eliminating the term with the highest degree. At this point, nosotros are left with a term of lower caste than the divisor, and so we finish. The quotient is the expression above the line, and the remainder is the expression at the lesser.

Usually, we write this concisely equally follows:

The conventions used when preforming long division this way regarding the placement of the terms of the polynomials vary. All the same, the technique is the same.

Example i: Polynomial Long Division with a Offset-Degree Divisor

Utilise polynomial partition to simplify .

Answer

In this example, nosotros expect a zero rest:

So the simplification is

A upshot of a zero residual is that we get a factorization. In the special example of a linear divisor, we get the following.

The Cistron Theorem

The polynomial is divisible past (with zero residuum) if and but if .

In other words, when is a nil of the polynomial.

So precisely when .

Example 2: The Factor Theorem and Long Sectionalization

By factoring, find all the solutions to , given that is a factor of .

Answer

Since is a cistron of this polynomial, we can use the factor theorem to conclude that is a zilch of the polynomial. We can use polynomial partitioning to detect the other factors.

And then and we tin factorize this quadratic, for example, by inspection: and therefore

The gene corresponds to aught , the factor gives the zip . So the zeros are

Using the same method, we can perform polynomial long partition when the divisor is of degree greater than ane. In the side by side instance, nosotros will demonstrate this.

Example 3: Polynomial Long Division with Higher-Degree Divisors

Employ polynomial long segmentation to find the quotient and the residue for , where and .

Answer

Applying the long segmentation algorithm, we get the following division:

Hence, the quotient and remainder .

Of class, we should not always expect the resulting polynomials and to have integer coefficients, fifty-fifty when and practice. The next instance demonstrates this.

Case 4: Polynomial Long Division

Express the sectionalization in the class .

Answer

Using the long division algorithm, we get the post-obit long division:

Hence,

The factor theorem is a special case of the balance theorem.

The Remainder Theorem

When the polynomial is divided past , the residuum is the constant .

Example 5: The Remainder Theorem

Find the balance when is divided past .

Answer

Although this can exist done by long sectionalization, we can also use the residuum theorem. Nosotros practice have to be careful almost the application, considering is not for any . Nevertheless, suppose that with residual the constant and caliber . Since we tin rewrite the above as

This says that the remainder when is divided by is the aforementioned as the residual on division past . Since this has the correct grade, the remainder theorem applies and

Example 6: Using Polynomial Long Division

Find the value of that makes the expression divisible past .

Answer

We tin can do this by polynomial division. We should expect a remainder of degree 1 or less which will involve the constant and setting that to zero volition determine the required .

The commencement pace is to ensure that the dividend is written correctly in descending powers of :

Using the algorithm:

nosotros find the residuum has caste 0 and is

Since is a factor only if the segmentation gives a zero residual, the condition on is that ; in other words

Find that the method used above will always work. An alternative (which is applicable here) is to use the residuum theorem. Observe that has zeros . If with some quotient , then evaluating at, say, should give zilch. Indeed, we discover

Central Points

• Using a similar algorithm for integer long division, we tin can divide polynomials.
• If we divide a polynomial by a factor, we go no remainder. Otherwise, we volition be left with a remainder of degree less than the degree of the divisor.
• For simple linear factors of the form , nosotros tin can find the residuum by applying the the remainder theorem which states that when the polynomial is divided by , the residuum is the abiding .

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