Chapter 1 · Algebraic Proofs · Concept 2
1-5 Quotient and Remainder of Polynomials
In lesson 1-4 you were given two polynomials and asked to divide them. Now the question is turned round: you are told the answer, and asked what was divided.
Guiding question
Why must the remainder always be smaller than the divider, and how does that let you check your own work?
Learning outcome. Distinguishes the quotient and the remainder in polynomial division, deduces that the degree of the remainder is less than the degree of the divisor, and uses this fact to verify the correctness of the division.
One line, and the whole lesson is in it
A = B × Q + R
If a polynomial A is divided by a polynomial B, giving a quotient Q and a remainder R, then that equality always holds.
You have always known this.
When 7 is divided by 3, the quotient is 2 and the remainder is 1.
7 = 3 × 2 + 1
Exactly the same shape. Polynomials do nothing new here.
When a question gives you the quotient and the remainder, put what you know into A = BQ + R and work out the rest.
Every question in this lesson is one of two kinds. They look different. They are not.
Find A
You are given B, Q and R.
A = B × Q + R
Just multiply out and add. No division at all.
Find B
You are given A, Q and R.
B = (A − R) ÷ Q
Take R off first, then divide. One division, and it must come out exactly.
The second line is just the first one rearranged. Take R from both sides, then divide both sides by Q. Nothing new was learned in between.
Choose what you are looking for, then fill in the three you were given. Every question on these two pages fits in here.
A = B × Q + R
- Multiply B by Q. (x² − 2x − 1)(x + 3) = x³ + x² − 7x − 3
- Add the remainder. x³ + x² − 7x − 3 + (−2x + 5)
- So A = x³ + x² − 9x + 2
1. When A is divided by x2 − 2x − 1, the quotient is x + 3 and the remainder is −2x + 5. Find A.
- 1
Write down which letter is which.
B = x2 − 2x − 1 · Q = x + 3 · R = −2x + 5Doing this first stops you putting the quotient where the divider goes.
2. When 4x3 + 8x − 9 is divided by B, the quotient is 2x − 3 and the remainder is x + 15. Find B.
- 1
Put what you know into the rule.
4x3 + 8x − 9 = B × (2x − 3) + (x + 15)This time B is the unknown one, so it stays as a letter for now.
Say you are dividing by B = x² + 1, whose highest power is 2. Move the slider to change how big the leftover is.
This is a real remainder.
Its highest power (1) is smaller than 2. There is no way to take another whole x² + 1 out of it, so the division has finished.
That is the whole answer to the Reflect question below: if the leftover is still big enough to divide, you have not finished dividing.
Work with a partner. Solve question 2 with your group, then talk about the answer. Decide first which of the two kinds each question is.
Find the polynomials A and B that satisfy these conditions.
When x4 + x2 + 3x − 1 is divided by B, the quotient is x2 + x + 1 with remainder 3x − 2. Find the polynomial B.
Stopping too early.
Ending the division before the leftover has a smaller highest power than the divider. If it is still big enough to divide, you have not finished.
The remainder does not have to be zero.
A remainder of −2x + 5 is a perfectly good answer. Expecting zero every time makes students go round again and spoil a correct division.
Taking away a negative remainder.
A − R when R = −4 means A + 4. This one step decides Try 2, and it is the easiest mark on the page to drop.
When you find B, the division must be exact.
If (A − R) ÷ Q leaves something over, you have made a slip. Use that as a free check — it tells you before the examiner does.
That is Concept 2 finished. The review quiz covers both lessons.
Keep reading — this lesson marks itself done automatically.