Expert II Dividing Polynomials and Graphing a polynomial division math 30-1 only Flashcards
(10 cards)
What are the numbers in a division problem called?
The dividend divided by the divisor gives the quotient plus a remainder to be divided by the divisor.
Make sure you understand every single word in the above sentence before moving on.
The remainder is the top number of the fraction (the numerator) in the actual answer. The remainder is 0 if the dividend was divisible by the divisor, and thus the quotient is the entire answer in that case.
Since
dividend / divisor = quotient + remainder / divisor
then
dividend = divisor x quotient + remainder
And later we will just call the dividend the polynomial (really the polynomial in the numerator) so
polynomial = divisor x quotient + remainder
P = DQ + R
Remember that the remainder is only the numerator! It is not the same thing as the portion left over after division, but it is the portion left to be divided so we can place a remainder on top of the divisor to show the actual portion remaining.
How do you divide a polynomial by a binomial? On this card we will explore old methods so that you have something to relate to.
Show that you know long division by stating the answer to 739 divided by 35. State your answer as an equation in two ways.
You can factor like you learned in Math 10C for any degree 2 trinomial using decomposition or even using the quadratic formula to find the roots and then list those roots back into the factored form.
However, we now need a better way to divide since sometimes we do not have a degree 2 trinomial for the dividend.
So for any polynomial to be divided by a binomial, we will look back to elementary school and long division. You should practice some of those first to remember how that works. Feel free to learn how to do three digit by two digit division on the abacus to give yourself a chance to visualize it better before proceeding with polynomial divisions.
Write 739 divided by 35 in your notebook and pay attention to your methods.
Dividing by a monomial when the remainder is zero will be much easier since this is just taking out a greatest common factor, and that was also learned in grade 10.
Show how to divide x² + 3x + 18 by x
To divide a polynomial by a binomial we must do this method:
Place the dividend under the house and the divisor to the left of this.
Ask yourself how many times the first term in the divisor can fit into the dividend. Then write the little bit of the quotient on top. Here x can fit into x² a certain amount of times, and the little quotient is x times. Place an x on top.
Now multiply the entire divisor (here that is just the x on the left) by the quotient you just found (here that is is x on top) and place that underneath your work so that you remember to subtract that value as the next step. So you should now have an x² under your house, but you may have two or more terms in the future if your divisor was a polynomial.
Then subtract the multiplication of all the terms in the divisor with that first term. So here we subtract x² and get 0.
Bring down the other numbers left to be divided. This should be obvious since we were just trying to subtract from our entire dividend earlier, so those values to the right still exist and need to be taken care of.
So now we are trying to fit x into 3x + 18. How many times does x fit into 3x? Three times is correct. Write + 3 as your small quotient on top of the house so that you now have x + 3 as an answer for the moment. Then subtract the product (3 times x) that you can make from the entire divisor (just x) and the small quotient you just made (that’s the three).
You should have 18 left to divide. How many times can x fit into 18 is a confusing question. What can you multiply 18 by to get to x? This is really just 18/x since (18/x)(x) = 18 but we just deal with this like the remainders in elementary school. We say we cannot fit x into 18, so 18 is the remainder left to be divided by the divisor, which is x in this case.
(x² + 3x + 18) / x = x + 3 + 18/x
Show how to divide x³ - 5x + 6 by x
Show how to divide 2x² - x - 18 by x + 2
Divide y³ - 6y + 20 by y - 3
Synthetic division
Divide 6x⁴ + 5x³ - 4x + 1 by x - 1
Use synthetic division to divide this problem that you have seen before with long division:
x³ - 5x + 6 divided by x
Synthetic division by a non-monial
What is 6x⁴ + 5x³ - 4x + 1 divided by 2x -1 ?
Show long division first, then do synthetic division making sure to divide the entire divisor by 2 before you start and divide the quotient by that same number right at the end. Essentially you are taking two steps to divide it, with two divisions created by the factors of the divisor.
where a polynomial divided by (x + 2) has a remainder of -239, what is the value of the polynomial when you sub in x = -2 in that polynomial?
Divide 5x⁵ -6x⁴ + 3x² - 2x + 1 by x - 2 to investigate this property.
It is also equal to -239
You will find that the remainder is equal to the value of the polynomial with that root subbed in.
