Polynomial Long Division | MathHelp.com
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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.
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