What is the Remainder Theorem Formula?
The remainder theorem formula applies to a polynomial P(x) that is divided by a linear binomial.
When dividing, the remainder can be calculated by performing the operations on the opposite of the constant in the binomial.
So, P(x) ÷ (x + a) = P(–a)
What is the Opposite of the Constant?
In the remainder theorem formula above, our binomial is (x + a) and our our constant is a, so the opposite of our constant is –a.
To see the free examples, please scroll to the sections below the quiz.
Remainder Theorem – Exercises
Instructions: Use the remainder theorem formula to find the remainder in the following problems.
[WpProQuiz 26]
Remainder Theorem – Example
The remainder theorem starts off with P(x), which means a polynomial that contains x as a variable.
The polynomial is divided by a linear binomial in the form (x + a), where the constant a can be either a positive or negative number.
Example: Divide the polynomial x3 + 3x2 – 4x – 10 by x + 3.
STEP 1:
Identify the binomial and the opposite of its constant.
In our example, our binomial is x + 3
The constant in the binomial is 3.
So, the opposite of the constant in the binomial is –3.
STEP 2:
Replace x in the polynomial with the opposite of the constant (which is –3 in this question) to find the remainder.
(x3 × 1) + (x2 × 1) + (x1 × –4) – 10
[(–3)3 × 1] + [(–3)2 × 1] + [(–3)1 × –4] + [–10] =
[(–3)3] + [(–3)2 × 3] + [(–3)1 × –4] + [–10] =
(–27) + [(9) × 3] + [(–3) × –4] – 10 =
–27 + 27 + 12 – 10 = 2
Check Your Result:
Then check this result by performing long division as shown below.
x2 – 4
x + 3) x3 + 3x2 – 4x – 10
–(x3 + 3x2)
– ( 0 – 4x – 10)
– ( –4x – 12)
2
You should also have a look at our posts on synthetic division, factoring, and quadratics.
Then try our free algebra test.
