I'll draw another horizontal "equals" bar, and change the signs on all the terms in the bottom row:. By design, the 10 x 's cancelled off. By happenstance, the 10 's cancelled off, too.
Then my answer, from across the top of the division symbol, is:. Since the remainder on the division above was zero that is, since there wasn't anything left over , the division "came out even". When you do regular division with numbers and the division "comes out even", it means that the number you divided by is a factor of the number you're dividing.
For instance, if you divide 50 by 10 , the answer will be a nice neat " 5 " with a zero remainder, because 10 is a factor of Any time you get a zero remainder, the divisor is a factor of the dividend. By the way, take note of how I figured out what to put on top of the long-division symbol in the exercise above: I divided the leading term of whatever I was dividing into by the leading term of what I was dividing by. Regardless of whether a particular division will have a non-zero remainder, this method will always give the right value for what you need on top.
In this way, polynomial long division is easier than numerical long division, where you had to guess-n-check to figure out what went on top. Let's do one more example with a division that comes out "even", so we can verify our result by doing the factorization and cancellation.
This fraction-reduction can be done in either of two ways: I can factor the quadratic and then cancel the common factor, like this:. But what if I didn't know how to factor or if I have to "show my work" for the long polynomial division on a test? As previously, I'll start the long division by working with the leading terms of the divisor and the dividend. The leading term of the dividend is x 2 and the leading term of the divisor is x.
Dividing x 2 by x gives me x , so that's what I put up on top, directly over the x 2 in the dividend:. Dividing the leading 2 x by the divisor's leading x gives me 2 , so that's what I put on top of the division symbol, right above the 9 x in the dividend:.
The divisor what you are dividing by goes on the outside of the box. The dividend what you are dividing into goes on the inside of the box. When you write out the dividend make sure that you write it in descending powers and you insert 0's for any missing terms. For example, if you had the problem , the polynomial , starts out with degree 4, then the next highest degree is 1.
It is missing degrees 3 and 2. So if we were to put it inside a division box we would write it like this:. This will allow you to line up like terms when you go through the problem. When you set this up using synthetic division write c for the divisor x - c. Then write the coefficients of the dividend to the right, across the top. Include any 0's that were inserted in for missing terms. Step 2: Bring down the leading coefficient to the bottom row.
Step 3: Multiply c by the value just written on the bottom row. Place this value right beneath the next coefficient in the dividend: Step 4: Add the column created in step 3. Write the sum in the bottom row: Step 5: Repeat until done. Step 6: Write out the answer. The numbers in the last row make up your coefficients of the quotient as well as the remainder.
The final value on the right is the remainder. Working right to left, the next number is your constant, the next is the coefficient for x , the next is the coefficient for x squared, etc The degree of the quotient is one less than the degree of the dividend.
For example, if the degree of the dividend is 4, then the degree of the quotient is 3. Example 1 : Divide using synthetic division:. Long division would look like this: Synthetic division would look like this: Step 2: Bring down the leading coefficient to the bottom row. Example 2 : Divide using synthetic division: Step 1: Set up the synthetic division. Remainder Theorem If the polynomial f x is divided by x - c , then the reminder is f c. This means that we can apply synthetic division and the last number on the right, which is the remainder, will tell us what the functional value of c is.
Example 3 : Given , use the Remainder Theorem to find f Lesson: 1a. Lesson: 1b. Lesson: 2. Lesson: 2a. Lesson: 2b. Lesson: 2c. Lesson: 3. Lesson: 4. Intro Learn Practice. Dividing Polynomials Sometimes, most often when dealing with rational expressions, it will be necessary to divide polynomials.
Steps in Solving Synthetic Division: Synthetic division is a "quick" process that allows one to more efficiently divide polynomials, compared to using good ol' fashioned long division. Setup the Synthetic Division. Add the Column Created in Stem 3. Repeat until Done. Write out the Final Answer. Find first term by dividing the first term of the numerator by the first term of the denominator,. Find second term by dividing the first term of the numerator by the first term of the denominator.
Find last term by dividing the first term of the numerator by the first term of the denominator. Find first term by dividing the first term of the numerator by the first term of the denominator. Do better in math today Get Started Now. What is a polynomial function? Polynomial long division 3. Polynomial synthetic division 4. Remainder theorem 5. Factor theorem 6.
Rational zero theorem 7. Characteristics of polynomial graphs 8.
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