Remainder Theorem and Factor Theorem (2024)

Or: how to avoid Polynomial Long Division when finding factors

Do you remember doing division in Arithmetic?

"7 divided by 2 equals 3 with a remainder of 1"

Each part of the division has names:

Which can be rewritten as a sum like this:

Polynomials

Well, we can also divide polynomials.

f(x) ÷ d(x) = q(x) with a remainder of r(x)

But it is better to write it as a sum like this:

Like in this example using Polynomial Long Division (the method we want to avoid):

Example: 2x²−5x−1 divided by x−3

f(x) is 2x²−5x−1
d(x) is x−3

After dividing we get the answer 2x+1, but there is a remainder of 2.

q(x) is 2x+1
r(x) is 2

In the style f(x) = d(x) q(x) + r(x) we can write:

2x²−5x−1 = (x−3)(2x+1) + 2

And there is a key feature:

The degree of r(x) is always less than d(x)

Say we divide by a polynomial of degree 1 (such as "x−3") the remainder will have degree 0 (in other words a constant, like "4").

We will use that idea in the "Remainder Theorem".

The Remainder Theorem

When we divide f(x) by the simple polynomial x−c we get:

f(x) = (x−c) q(x) + r(x)

x−c is degree 1, so r(x) must have degree 0, so it is just some constant r:

Example: The remainder after 2x²−5x−1 is divided by x−3

(Our example from above)

We don't need to divide by (x−3) ... just calculate f(3):

2(3)²−5(3)−1 = 2x9−5x3−1
= 18−15−1
= 2

And that is the remainder we got from our calculations above.

We didn't need to do Long Division at all!

Example: The remainder after 2x²−5x−1 is divided by x−5

Similar to our example above but this time we divide by "x−5"

"c" is 5, so let us check f(5):

2(5)²−5(5)−1 = 2x25−5x5−1
= 50−25−1
= 24

The remainder is 24

Once again ... We didn't need to do Long Division to find that.

The Factor Theorem

Now ...

What if we calculate f(c) and it is 0?

... that means the remainder is 0, and ...

... (x−c) must be a factor of the polynomial!

We see this when dividing whole numbers. For example 60 ÷ 20 = 3 with no remainder. So 20 must be a factor of 60.

Example: x²−3x−4

f(4) = (4)²−3(4)−4 = 16−12−4 = 0

so (x−4) must be a factor of x²−3x−4

And so we have:

The Factor Theorem:

When f(c)=0 then x−c is a factor of f(x)

And the other way around, too:

When x−c is a factor of f(x) then f(c)=0

Why Is This Useful?

Knowing that x−c is a factor is the same as knowing that c is a root (and vice versa).

The factor "x−c" and the root "c" are the same thing

Know one and we know the other

For one thing, it means that we can quickly check if (x−c) is a factor of the polynomial.

Example: Find the factors of 2x³−x²−7x+2

The polynomial is degree 3, and could be difficult to solve. So let us plot it first:

The curve crosses the x-axis at three points, and one of them might be at 2. We can check easily:

f(2) = 2(2)³−(2)²−7(2)+2
= 16−4−14+2
= 0

Yes! f(2)=0, so we have found a root and a factor.

So (x−2) must be a factor of 2x³−x²−7x+2

How about where it crosses near −1.8?

f(−1.8) = 2(−1.8)³−(−1.8)²−7(−1.8)+2
= −11.664−3.24+12.6+2
= −0.304

No, (x+1.8) is not a factor. We could try some other values near by and maybe get lucky.

But at least we know (x−2) is a factor.

Let us check using Polynomial Long Division:

2x²+3x−1
x−2)2x³− x²−7x+2
2x³−4x²
3x²−7x
3x²−6x
−x+2
−x+2
0

As expected the remainder is zero.

Better still, we are left with the quadratic equation 2x²+3x−1 which is easy to solve:

It's roots are −1.78... and 0.28..., so the final result is:

2x³−x²−7x+2 = (x−2)(x+1.78...)(x−0.28...)

We were able to solve a difficult polynomial.

Summary

The Remainder Theorem:

When we divide a polynomial f(x) by x−c the remainder is f(c)

The Factor Theorem:

When f(c)=0 then x−c is a factor of f(x)
When x−c is a factor of f(x) then f(c)=0

482, 483, 4014, 4015, 484, 485, 4016, 1124, 1125, 4017

Challenging Questions:

96,227,228,229,230,231

Polynomial Long Division Algebra Index

FAQs

Remainder Theorem and Factor Theorem? ›

The remainder theorem tells us that for any polynomial f(x) , if you divide it by the binomial

binomial

Binomial is a polynomial with only terms. For example, x + 2 is a binomial, where x and 2 are two separate terms. Also, the coefficient of x is 1, the exponent of x is 1 and 2 is the constant here. Therefore, A binomial is a two-term algebraic expression that contains variable, coefficient, exponents and constant.

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Definition, Operations on Binomials & Examples - BYJU'S

x−a , the remainder is equal to the value of f(a) . The factor theorem tells us that if a is a zero of a polynomial f(x) , then (x−a) is a factorof f(x) , and vice-versa.

Read On ›

What is the difference between factor theorem and remainder theorem? ›

Basically, the remainder theorem links remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its zeros.

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What is the remainder theorem and factor theorem formula? ›

The Factor and Remainder Theorems

If p(x) is a polynomial of degree 1 or greater and c is a real number, then when p(x) is divided by x−c, the remainder is p(c). If x−c is a factor of the polynomial p, then p(x)=(x−c)q(x) for some polynomial q.

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What is the remainder theorem? ›

The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial , x - a, the remainder of that division will be equivalent to f(a).

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What is the factor theorem in simple terms? ›

In mathematics, factor theorem is used when factoring the polynomials completely. It is a theorem that links factors and zeros of the polynomial. According to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and 'a' is any real number, then, (x-a) is a factor of f(x), if f(a)=0.

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What is the conclusion of the remainder and factor theorem? ›

We can conclude, then, that if h is a zero of f(x), then (x - h) is a factor of f(x). Similarly, if (x - h) is a factor of f(x), then the remainder of the Division Algorithm f(x) = (x-h) q(x) + r is 0. This tells us h is a zero. This pair of implications is known as the Factor Theorem.

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How do you know if the remainder is a factor? ›

Learn how to determine if an expression is a factor of a polynomial by dividing the polynomial by the expression. If the remainder is zero, the expression is a factor.

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What is factor and remainder theorem IB math? ›

The Remainder Theorem states that when a polynomial P(x) is divided by x - a, the remainder is P(a). Now, if P(a) equals 0, it means that the polynomial is perfectly divisible by x - a without any remainder. This is the essence of the Factor Theorem, which states that if P(a) equals 0, then x - a is a factor of P(x).

What is the advantage of the remainder theorem? ›

The advantage of the remainder theorem is that we can determine if a value is a factor by checking if the remainder is zero. The Factor Theorem is based upon the properties of the Remainder Theorem. If f(a)=0 then the remainder is 0 and x-a is a factor.

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What is the remainder theorem calculator? ›

Remainder Theorem Calculator is a free online tool that displays the quotient and remainder of division for the given polynomial expressions. BYJU'S online remainder theorem calculator tool makes the calculation faster, and it displays the result in a fraction of seconds.

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What is the formula for the remainder and factor theorem? ›

The remainder theorem states that the remainder when p(x) is divided by (x - a) is p(a). The factor theorem states that (x - a) is a factor of p(x) if and only if p(a) = 0. It is used to find the remainder. It is used to decide whether a linear polynomial is a factor of the given polynomial or not.

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What is the application of remainder theorem in real life? ›

Real-life Applications

The remainder theorem provides a more efficient avenue for testing whether certain numbers are roots of polynomials. This theorem can increase efficiency when applying other polynomial tests, like the rational roots test.

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What is the factor formula? ›

Factoring formulas are used to write an algebraic expression as the product of two or more expressions. Some important factoring formulas are given as, (a + b)² = a² + 2ab + b. (a - b)² = a² - 2ab + b.

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What is the factor theorem and remainder theorem of synthetic division? ›

Also, the Remainder Theorem states that the remainder that we end up with when synthetic division is applied actually gives us the functional value. Another use is finding factors and zeros. The Factor Theorem states that if the functional value is 0 at some value c, then x - c is a factor and c is a zero.

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What is the difference between the remainder theorem and the division algorithm? ›

In summary, the division algorithm is a general method for dividing one polynomial by another, giving a quotient and a remainder, whereas the remainder theorem is a specific result that relates the remainder of dividing a polynomial by x - a to the evaluation of the polynomial at a.

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