Easily Factorizing Higher Degree Polynomials Using the Remainder Theorem and Factor Theorem

When dealing with polynomials of degree 3 or higher, it can often be challenging to factorize using traditional methods. However, there are two helpful theorems that can provide a shortcut: the Remainder Theorem and the Factor Theorem. These theorems are useful in finding the remainder and factors of complex polynomials. Before we dive into these theorems, let's first review the standard methods for dividing polynomials.

Division Algorithm

Similar to dividing numbers, we can express a dividend as a product of its divisor and quotient, plus a remainder. This is known as the Division Algorithm, which can be written as:

Dividend = (Divisor x Quotient) + Remainder

For example, when dividing 250 by 7, we get a quotient of 35 and a remainder of 5, which can be expressed as:

250 = 7 x 35 + 5

If the remainder is 0, then the divisor is a factor of the dividend, which can be written as:

Dividend = Factor x Quotient

Using this notation, we can say that:

250 = 5 x 50

These concepts and notation also apply to polynomials.

Long Division

Let's look at an example:

Polynomial: 4x^2-3x+6

Divisor: x-1

Using long division, we get:

Quotient: 4x+1

Remainder: 7

Therefore, we can express the polynomial as:

4x^2-3x+6 = (x-1)(4x+1) + 7

Synthetic Division

Another method for dividing polynomials is synthetic division. Let's use the same example as above:

Polynomial: 4x^2-3x+6

Divisor: x-1

Using synthetic division, we also get a remainder of 7. This method can be helpful to quickly divide higher degree polynomials.

The Remainder Theorem

The Remainder Theorem is used to find the remainder of a polynomial when divided by a linear polynomial, which is a first-degree polynomial in the form of g(x) = ax + b.

The Remainder Theorem is stated as:

If p is a polynomial divided by (x-a), then the remainder is p(a).

In general, this can be expressed as:

p(x) = (x-a)q(x) + p(a)

Where p is the dividend, (x-a) is the divisor, q is the quotient, and p(a) is the remainder.

Proof of the Remainder Theorem

Let p be a polynomial divided by (x-a), where a is a real number. The division algorithm can be written as:

p(x) = (x-a)q(x) + r(x)

When we plug in x=a, we get:

p(a) = (a-a)q(a) + r(a)

Therefore, the remainder, r, is equal to p(a).

Using the Remainder Theorem

Now, let's use the Remainder Theorem on our previous example:

Polynomial: 4x^2-3x+6

Divisor: x-1

Using the Remainder Theorem, we can say that the remainder is f(1). Evaluating this yields:

f(1) = 7

We get the same result as we did with long division and synthetic division. This shows the efficiency of using the Remainder Theorem to find the remainder of a polynomial.

The Remainder Theorem can also be applied to polynomials of higher degrees:

Example 1:

Polynomial: x^3+12x^2-3x+5

Divisor: x+3

Using the Remainder Theorem, we can find the remainder to be:

f(-3) = 95

Using long division, we also get a remainder of 95. This shows that the Remainder Theorem is applicable even for higher degree polynomials.

Example 2:

Polynomial: x^5-3x^3+7x^2-5x+2

Divisor: x-1

Using the Remainder Theorem, we can find the remainder to be:

f(1) = 2

Again, using long division, we get the same remainder of 2. This demonstrates the effectiveness of using the Remainder Theorem to find the remainder of a polynomial.

The Factor Theorem

The Factor Theorem is a formula used to completely factor a polynomial into a product of n factors, where n is the number of factors the polynomial has. This allows us to find the solutions to the equation given by the polynomial equal to zero.

The Factor Theorem and Its Proof

The Factor Theorem is used to obtain the roots of a polynomial by applying the Zero Product Property from the topic of Factoring Polynomials. The theorem states that a polynomial, p, has a factor (x – a) if and only if the remainder when p is divided by (x – a) is equal to 0. This can be expressed as p(x) = (x – a)q(x), where p(x) is the dividend, (x – a) is the factor, and q(x) is the quotient.

Proof of the Factor Theorem:

Let p be a polynomial divided by (x – a), where a is a real number. If (x – a) is a factor of p, then p(x) = (x – a)q(x). Plugging in x = a, we get p(a) = (a – a)q(x) ⇒ p(a) = 0 · q(x) ⇒ p(a) = 0. This shows that a is a root of p.

Conversely, if a is a root of p, then p(a) = 0. Applying the Remainder Theorem, we know that p(x) = (x – a)q(x) + p(a). Substituting p(a) = 0 gives us p(x) = (x – a)q(x) + 0 ⇒ p(x) = (x – a)q(x). Therefore, (x – a) is indeed a factor of p(x), proving the Factor Theorem.

Example: Using the Factor Theorem

Let us use the Factor Theorem to determine if x - 1 is a factor of the polynomial f(x) = 2x2 – 3x + 1. By the theorem, if x - 1 is a factor, then the remainder when f(x) is divided by x - 1 must be 0. Evaluating f(1), we get f(1) = 2(1)2 – 3(1) + 1 ⇒ f(1) = 2 – 3 + 1 ⇒ f(1) = 0. Hence, x - 1 is a factor of f(x). This showcases the usefulness of the Factor Theorem in completely factoring a polynomial and finding its roots.

Applying the Factor Theorem to Polynomials of Degree Greater than Two

The Factor Theorem can also be used for polynomials with a degree greater than two. Let us explore two examples to demonstrate this concept.

Example 1: Using the Factor Theorem and Synthetic Division

Using the Factor Theorem, we can determine if the binomial x + 2 is a factor of the cubic polynomial f(x) = x^3 - 4x^2 - 7x + 10. Then, we can confirm our result using synthetic division.

According to the Factor Theorem, if x + 2 is a factor of f(x), then the remainder when f(x) is divided by x + 2 will be 0. Substituting x = -2, we get f(-2) = (-2)^3 - 4(-2)^2 - 7(-2) + 10 ⇒ f(-2) = -8 - 4(4) + 14 + 10 ⇒ f(-2) = 0. Therefore, we can conclude that x + 2 is a factor of f(x). By using synthetic division, we obtain a remainder of 0, confirming our result.

Example 2: Using the Factor Theorem and Synthetic Division to Verify a Result

We can also apply the Factor Theorem to determine if the binomial x - 1 is a factor of the cubic polynomial f(x) = 3x^3 - 11x^2 + 5x + 3. Then, we can use synthetic division to verify our result.

If x - 1 is a factor of f(x), then the remainder when f(x) is divided by x - 1 will be 0. Evaluating f(1), we get f(1) = 3(1)^3 - 11(1)^2 + 5(1) + 3 ⇒ f(1) = 3 - 11 + 5 + 3 ⇒ f(1) = 0. Therefore, we can conclude that x - 1 is a factor of f(x). Using synthetic division once again, we obtain a remainder of 0, verifying our result.

Finding Solutions of Polynomials Using the Factor Theorem

Using the Factor Theorem, we can fully factorize a polynomial, which can assist in finding solutions to the polynomial. We can use synthetic division or long division to determine the quotient associated with the polynomial's factor.

Solving a Cubic Polynomial

Let's solve the cubic polynomial f(x) = x^3 - 4x^2 - 7x + 10 = 0 as an example.

We know from the previous example that x + 2 is a factor of f. Using synthetic division, we find that the quotient associated with this divisor is the polynomial x^2 - 6x + 5. This means that the polynomial can be factorized as f(x) = (x + 2)(x^2 - 6x + 5) = 0. We can further factorize the quadratic trinomial x^2 - 6x + 5 using factoring techniques. This gives us the factorized form f(x) = (x + 2)(x - 1)(x - 5) = 0. Using the Zero Product Property, we get x + 2 = 0, x - 1 = 0, and x - 5 = 0. Solving for x, we get three solutions: x = -2, x = 1, and x = 5.

In order to find solutions to the polynomial f(x) = 3x^3 - 11x^2 + 5x + 3 = 0, we use the same approach. From the previous example, we know that x - 1 is a factor of f. Using synthetic division, we find that the quotient associated with this divisor is the polynomial 3x^2 - 8x - 3. This means that the polynomial can be factorized as f(x) = (x - 1)(3x^2 - 8x - 3) = 0. We can further factorize the quadratic trinomial 3x^2 - 8x - 3 to find the solutions to the polynomial.

The Complete Factorized Form

The factorized form of a polynomial, f(x), can be written as f(x) = (x - a)(ax + b)(x - c) = 0. This indicates that there are three possible roots for this polynomial: a, -b/a, and c.

The Remainder and Factor Theorems for Divisors of the Form (ax - b)

We have only considered divisors of the form (x - a) so far. However, there is another linear form of divisors: (ax - b). To understand the relationship between these divisors and the Remainder and Factor Theorems, we can use the standard formula below.

The Remainder and Factor Theorems for Divisor (ax - b)

The Remainder Theorem states that if a polynomial, p(x), is divided by (ax - b), then the remainder will be equal to p(b/a). Additionally, if p(b/a) = 0, then (ax - b) is a factor of p(x).

Let's use the polynomial f(x) = 3x^3 - 11x^2 + 5x + 3 as an example. In its completely factorized form, we can see that (3x + 1) is one of the factors. We can use the Factor Theorem to prove this by setting the remainder, f(-1/3), equal to zero.

Returning to the same example, we can use the Remainder Theorem to find the remainder of f(x) when divided by (5x - 7). By setting the remainder, f(7/5), equal to zero, we can calculate the remainder to be f(7/5) = -4161/25.

Remainder Theorem vs. Factor Theorem

There are some key differences between the Remainder and Factor Theorems. See the table below for a comparison.

• Remainder Theorem: associates the remainder of division by a binomial with the value of a function at a point. Used to find the remainder of a polynomial when divided by a linear polynomial.Understanding the Factor and Remainder Theorems for Polynomials
• The Factor and Remainder Theorems are valuable concepts in the study of polynomials. These theorems allow us to factor polynomials and find their remainders when dividing by linear polynomials, respectively.
• Explaining the Factor and Remainder Theorems
• The Factor Theorem is a technique that enables us to factor a polynomial into n factors, whereas the Remainder Theorem is a formula for finding the remainder of a polynomial divided by a linear polynomial.
• How to Use the Factor and Remainder Theorems
• If given a polynomial, f(x), and a linear equation, x - c, we can apply the Remainder Theorem by setting the value of f(c) as the remainder. To determine if x - c is a factor of f(x), we can utilize the Factor Theorem by setting f(c) equal to zero.
• Proof of the Factor and Remainder Theorems
• These theorems can be proven using the Division Algorithm.
• Dividing Polynomials with the Factor and Remainder Theorems
• To divide polynomials using these theorems, we have the option of using long division or synthetic division methods.