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Divide polynomials by linear binomials with remainders
By Khan Academy
Sal explains what polynomial long division is, and gives various examples of polynomial long divisions.
Divide polynomials by polynomials with remainders
By Khan Academy
Sal explains what polynomial long division is, and gives various examples of polynomial long divisions.
Divide polynomials by monomials with remainders
By Khan Academy
Sal explains what polynomial long division is, and gives various examples of polynomial long divisions.
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!
05 - Polynomial Long Division - Part 1 (Division of Polynomials Explained)
By Math and Science
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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!
Synthetic Division - A Shortcut for Long Division!
By PatrickJMT
This video demonstrates how to perform synthetic division of polynomials.
Polynomial Long Division | MathHelp.com
By MathHelp.com
McCauley can paint a house in 10 hours, while it takes Clayton 15 hours. If they work together, how long will it take them to paint the house? To solve this kind of a problem, which is called a work problem, it’s important to understand the following idea. Since McCauley can paint a house in 10 hours, we know that in 1 hour, McCauley can paint 1/10 of the house. And in 2 hours, McCauley can paint 2/10 of the house. Therefore, in t hours, McCauley can paint t tenths of the house. And since it takes Clayton 15 hours to paint the house, in t hours, Clayton can paint t fifteenths of the house. Pause the audio for a moment if you need time to understand this idea…Now, to solve the problem, we use the following formula: part of job done by McCauley + part of job done by Clayton = 1 job done. And we’re asked how long will it take them to paint the house, so we’re looking for the time, or t. Remember that in t hours, McCauley can paint t/10 of the house, so the part of the job done by McCauley is t over 10. And in t hours, Clayton can paint t/15 of the house, so the part of the job done by Clayton is t over 15. Now, we have the equation t/10 + t/15 = 1. To solve this equation for t, we first get rid of the fractions by multiplying both sides of the equation by the common denominator of 10 and 15, which is 30. Distributing on the left side, 30 times t over 10 is 30t over 10, which simplifies to 3t, and 30 times positive t over 15 is positive 30t over 15, which simplifies to positive 2t. And on the right, 1 times 30 is 30. So we have 3t + 2t = 30, or 5t = 30, and dividing both sides by 5, t = 6. So if Clayton and McCauley work together, they can paint the house in 6 hours. Finally, it’s a good idea to check your answer. If they work together for 6 hours, then McCauley paints 6/10 of the house, and Clayton paints 6/15 of the house, so we have 6/10 + 6/15 = 1. And reducing on the left side, we have 3/5 + 2/5 = 1, which simplifies to 5/5 = 1, which is a true statement, so our answer checks
Finding all the Zeros of a Polynomial - Example 2 - YouTube
By PatrickJMT
Finding all the Zeros of a Polynomial - Example 2. In this video, I use the rational roots test to find all possible rational roots; after finding one I use long division to factor, and then repeat! Very fun!
What Does the Fundamental Theorem of Algebra Tell Us about a Function?
By AutenA2Math
This video explains the Fundamental Theorem of Algebra and how it can assist us when solving a polynomial equation. (For those not familiar with the technique, this video demonstrates the use of "synthetic division.") Remember that the Fundamental Theorem of Algebra only applies when working in the Complex Number set. A polynomial like x^2 + 1 has no real roots, but it does have two complex roots.
Evaluating Logarithms | MathHelp.com
By MathHelp.com
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.
12 - The Factor Theorem, Part 1 (Factoring Polynomials in Algebra)
By Math and Science
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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.
18 - Descartes Rule of Signs, Part 1 (Find Roots of Polynomials)
By Math and Science
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