on April 27, 2022

What is quotient of polynomial?

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Polynomial Division De-Mystified: Finding the “Answer” (aka the Quotient)

Polynomials. They might sound intimidating, but they’re really just the basic building blocks of so much math. Think of them as the LEGO bricks of equations, popping up everywhere from simple algebra problems to the simulations scientists use to model the world. And just like with regular numbers, you can do all sorts of things with polynomials – add them, subtract them, multiply them, and, yes, divide them! When you dive into polynomial division, you end up with two main results: the quotient and, sometimes, a remainder. Let’s break down what that quotient thing actually is.

Polynomial Division: A Quick Refresher

Okay, so polynomial division is basically what it sounds like: dividing one polynomial by another. The thing you’re dividing into is called the dividend (think of it as the thing getting split up), and the thing you’re dividing by is the divisor (the thing doing the splitting). The whole process gives you two important pieces:

  • The Quotient: This is the main star of our show! It’s basically the “answer” you get from the division. It tells you how many times the divisor goes into the dividend.
  • The Remainder: Sometimes, the divisor doesn’t fit perfectly into the dividend. The remainder is what’s left over – the bit that doesn’t quite divide evenly.

You can think of it like this: if you’re splitting a pizza (the dividend) among friends (the divisor), the quotient is how many slices each person gets, and the remainder is any leftover slices. The relationship looks like this:

Dividend = (Divisor × Quotient) + Remainder

So, What Is the Quotient, Really?

In simple terms, the quotient is the “answer” to your division problem. It’s the polynomial you end up with after you’ve done all the dividing. To be a bit more technical, it’s the factor that, when multiplied by the divisor, gets you the dividend (or gets you really close, if there’s a remainder).

How Do You Actually Find the Quotient?

There are a couple of main ways to tackle polynomial division and snag that quotient:

  • Long Division: Remember doing long division with numbers back in school? Well, this is the same idea, just with polynomials! It works for dividing any two polynomials, no matter how complicated. It’s a step-by-step process where you divide, multiply, and subtract until you get to the quotient and the remainder.

    • First, get everything lined up neatly. Write both polynomials in order, from the highest power of x down to the lowest. If you’re missing any terms (like if you have an x3 but no x2), stick in a zero as a placeholder.
    • Then, divide the first term of the dividend by the first term of the divisor. That’s the first term of your quotient!
    • Multiply the whole divisor by that first term of the quotient, and subtract the result from the dividend.
    • Bring down the next term from the dividend, and repeat the whole process. Keep going until the “leftover” bit has a smaller power of x than the divisor. That leftover bit is your remainder, and everything you wrote down at the top is your quotient!

  • Synthetic Division: This is like a shortcut, but it only works when you’re dividing by something simple, like (x – a), where ‘a’ is just a number. It’s much faster than long division when you can use it.

    • Write down all the numbers in front of the ‘x’ terms in the dividend.
    • Write down the value of ‘a’ (from x-a) to the left.
    • Bring down the first number.
    • Multiply that number by ‘a’, and write the result under the next number.
    • Add those two numbers together.
    • Repeat the multiply-and-add steps until you’ve used all the numbers. The last number you get is the remainder, and the other numbers are the coefficients of the quotient. Just remember that the highest power of x in the quotient will be one less than what you started with in the dividend.

    • Example Time!

    Let’s say we want to divide (x2 – 1) by (x – 1) using long division. Here’s how it looks:

    basic

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