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Evaluating Logarithms | MathHelp.com

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In this example, notice that we have a polynomial divided by a binomial, and our binomial is in the form of an x term minus a constant term, or x – c. In this situation, instead of having to use long division, like we did in the previous lesson, we can divide the polynomials using synthetic division, which is a much more efficient method. Here’s how it works. We start by finding the value of c. Since –c = -3, we know that c = 3. Next, we put the value of c inside a box, so we put the 3 inside a box. It’s very important to understand that the number that goes inside the box always uses the opposite sign as the constant term in the binomial. In other words, since the constant term in the binomial is -3, the number that goes inside the box, is positive 3. Next, we write the coefficients of the dividend, which are 2, -7, 4, and 5. Be very careful with your signs. Now, we’re ready to start our synthetic division. First, we bring down the 2. Next, we multiply the 3 in the box times 2 to get 6, and we put the 6 under the -7. Next, we add -7 + 6 to get -1. Next, we multiply the 3 in the box times -1 to get -3, and we put the -3 under the 4. Next, we add 4 + -3 to get 1. Next, we multiply the 3 in the box times 1 to get 3, and we put the 3 under the 5. Finally, we add 5 + 3 to get 8. Now, notice that we have a 2, -1, 1, and 8 in the bottom row of our synthetic division. These values will give us our answer: the first 3 numbers represent the coefficients of the quotient, and the last number is the remainder. And it’s important to understand that our answer will be one degree less than the dividend. In other words, since our dividend starts with x cubed, and we’re dividing by x, our answer will start with x squared. So our answer is 2x squared – 1x + 1 + 8 over x – 3. Notice that we always use descending order of powers in our quotient. In this case x squared, x, and the constant. Finally, remember that we add the remainder over the divisor, just like we did in the previous lesson on long division, and we have our answer. It’s important to understand that we’ll get the same answer whether we use synthetic division or long division. However, synthetic division is much faster.

Synthetic Division | MathHelp.com

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08 - Learn Synthetic Division of Polynomials - Part 1

By Math and Science

Quality Math And Science Videos that feature step-by-step example problems!

10 - The Remainder Theorem of Synthetic Division & Polynomial Long Division - Part 1

By Math and Science

Quality Math And Science Videos that feature step-by-step example problems!

Synthetic Division | MathHelp.com

By MathHelp.com

In this example, it’s tempting to divide x squared + 5x – 6 by x + 1 by first factoring x squared + 5x – 6. The factors of -6 that add to positive 5 are +6 and -1, so we have x + 6 times x – 1 over x + 1. Notice, however, that nothing cancels. In this situation, we need a different method of dividing the polynomials, so we use long division. In other words, we rewrite x squared + 5x – 6 divided by x + 1 as x + 1 divided into x squared + 5x – 6. Now, our first step in the long division is to determine how many times x goes into x squared. Since x goes into x squared x times, we write an x above the x squared, just like we do with regular long division. Next, we multiply the x times the x + 1 in the divisor to get x squared + x, and we write the x squared + x underneath the x squared + 5x. Next, we subtract x squared + x from x squared + 5x. And watch out for this step: it’s an area where most of the common mistakes in these types of problems are made. Instead of subtracting, I would change the sign of each term in x squared + x, so we have negative x squared + negative x, then add the columns. So we have x squared + negative x squared, which cancels out, and positive 5x + negative x, which is positive 4x. Next, we bring down the -6, in regular long division. Now, we need to determine how many times x goes into 4x. Since x goes into 4x 4 times, we write a positive 4 in our answer. Next, we multiply positive 4 times x + 1 to get 4x + 4, and we write the 4x + 4 underneath the 4x – 6. Next, we subtract 4x + 4 from 4x – 6. In other words, we change the signs on 4x + 4 to -4x + -4, and we add. 4x + -4x cancels out, and -6 + -4 is -10. And since there are no other numbers to bring down, we have a remainder of -10. Finally, remember from the previous example that we add the remainder over the divisor to the quotient. In other words, we add -10 over x + 1 to x + 4, and we have x + 4 + -10 over x + 1. So x squared + 5x – 6 divided by x + 1 simplifies to x + 4 + -10 over x + 1.