Factoring Polynomials Using Synthetic Division A Step By Step Guide

Jul 16, 2025 by Jeany 68 views

In the realm of mathematics, specifically algebra, factoring polynomials is a fundamental skill. It involves breaking down a polynomial expression into simpler factors that, when multiplied together, yield the original polynomial. Factoring polynomials is essential for solving equations, simplifying expressions, and understanding the behavior of functions. Among the various techniques available for factoring polynomials, synthetic division stands out as an efficient and elegant method, particularly when dealing with polynomials of higher degrees. Synthetic division provides a streamlined approach to dividing a polynomial by a linear factor, which in turn helps us identify roots and ultimately factor the polynomial completely.

In this comprehensive guide, we will delve into the intricacies of synthetic division and explore its application in factoring polynomials. We will begin by laying the groundwork, defining key concepts such as polynomials, factors, and roots. Then, we will embark on a step-by-step journey through the synthetic division process, illustrating each step with clear examples. Furthermore, we will address the crucial role of the Remainder Theorem and the Factor Theorem in synthetic division. Finally, we will tackle the given problem, $x^4 + 6x^3 + 33x^2 + 150x + 200$ , employing synthetic division to determine its factored form. By the end of this exploration, you will have a solid understanding of synthetic division and its power in factoring polynomials.

Understanding Polynomials, Factors, and Roots

Before we dive into synthetic division, let's establish a firm understanding of the core concepts involved: polynomials, factors, and roots.

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Polynomials can be classified based on their degree, which is the highest power of the variable in the expression. For example, $x^2 + 3x + 2$ is a quadratic polynomial (degree 2), while $x^3 - 4x^2 + 5x - 2$ is a cubic polynomial (degree 3).

A factor of a polynomial is an expression that divides the polynomial evenly, leaving no remainder. In other words, if we can write a polynomial $P(x)$ as a product of two expressions, say $A(x)$ and $B(x)$ , then $A(x)$ and $B(x)$ are factors of $P(x)$ . For instance, $(x + 1)$ and $(x + 2)$ are factors of the quadratic polynomial $x^2 + 3x + 2$ , since $(x + 1)(x + 2) = x^2 + 3x + 2$ .

A root of a polynomial is a value of the variable that makes the polynomial equal to zero. Roots are also known as zeros of the polynomial. For example, the roots of the quadratic polynomial $x^2 + 3x + 2$ are -1 and -2, because $(-1)^2 + 3(-1) + 2 = 0$ and $(-2)^2 + 3(-2) + 2 = 0$ . Roots play a crucial role in factoring polynomials, as each root corresponds to a linear factor of the polynomial. If $r$ is a root of a polynomial $P(x)$ , then $(x - r)$ is a factor of $P(x)$ .

Synthetic Division: A Step-by-Step Guide

Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form $(x - c)$ , where $c$ is a constant. It provides a more efficient alternative to long division, especially when dealing with polynomials of higher degrees. Let's break down the synthetic division process into a series of steps:

Write down the coefficients: Begin by writing down the coefficients of the polynomial in descending order of powers of the variable. If any powers are missing, include a coefficient of 0 as a placeholder. For example, if we are dividing $x^3 - 4x + 6$ by $(x - 2)$ , we would write down the coefficients as 1, 0, -4, and 6, noting the 0 coefficient for the missing $x^2$ term.
Identify the divisor's root: Determine the value of $c$ from the linear factor $(x - c)$ . This value is the root of the divisor. In our example, the divisor is $(x - 2)$ , so $c = 2$ .
Set up the synthetic division table: Draw a horizontal line and a vertical line to create a table-like structure. Write the value of $c$ (the root) to the left of the vertical line. Write the coefficients of the polynomial to the right of the vertical line, above the horizontal line.
Bring down the first coefficient: Bring down the first coefficient of the polynomial below the horizontal line. This coefficient becomes the first entry in the quotient row.
Multiply and add: Multiply the value of $c$ (the root) by the first entry in the quotient row. Write the result below the next coefficient of the polynomial. Add the two numbers in that column and write the sum below the horizontal line. This sum becomes the next entry in the quotient row.
Repeat: Repeat step 5 for the remaining coefficients. Multiply the value of $c$ by the latest entry in the quotient row, write the result below the next coefficient, and add the two numbers in the column.
Interpret the results: The last number below the horizontal line is the remainder. The other numbers in the quotient row are the coefficients of the quotient polynomial, which has a degree one less than the original polynomial. If the remainder is 0, then $(x - c)$ is a factor of the polynomial.

To illustrate the process, let's perform synthetic division on $x^3 - 4x + 6$ divided by $(x - 2)$ :

2 | 1  0 -4  6
  |    2  4  0
  ------------
    1  2  0  6

The quotient is $x^2 + 2x$ , and the remainder is 6. Since the remainder is not 0, $(x - 2)$ is not a factor of $x^3 - 4x + 6$ .

The Remainder Theorem and the Factor Theorem

The Remainder Theorem and the Factor Theorem are two fundamental theorems that play a crucial role in synthetic division and polynomial factorization.

Remainder Theorem

The Remainder Theorem states that when a polynomial $P(x)$ is divided by $(x - c)$ , the remainder is equal to $P(c)$ . In other words, the remainder obtained from synthetic division is the same as the value of the polynomial evaluated at $x = c$ .

This theorem provides a powerful tool for evaluating polynomials. Instead of directly substituting a value for $x$ , we can use synthetic division to find the remainder, which is equal to the polynomial's value at that point. This can be particularly useful when dealing with polynomials of higher degrees or complex values of $x$ .

Factor Theorem

The Factor Theorem is a direct consequence of the Remainder Theorem. It states that $(x - c)$ is a factor of a polynomial $P(x)$ if and only if $P(c) = 0$ . In other words, $(x - c)$ is a factor of $P(x)$ if and only if $c$ is a root of $P(x)$ .

This theorem provides a powerful tool for identifying factors of polynomials. If we can find a value $c$ such that $P(c) = 0$ , then we know that $(x - c)$ is a factor of $P(x)$ . This is where synthetic division becomes invaluable. By performing synthetic division with a potential root $c$ , we can quickly determine if the remainder is 0. If the remainder is 0, then $(x - c)$ is a factor, and we have successfully factored the polynomial.

Factoring $x^4 + 6x^3 + 33x^2 + 150x + 200$ Using Synthetic Division

Now, let's apply our knowledge of synthetic division to factor the polynomial $x^4 + 6x^3 + 33x^2 + 150x + 200$ . Our goal is to find the factored form of this polynomial, which means expressing it as a product of simpler polynomials.

To begin, we need to identify potential roots of the polynomial. The Rational Root Theorem provides a guide for this. It states that if a polynomial has integer coefficients, then any rational root must be of the form $\frac{p}{q}$ , where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient. In our case, the constant term is 200 and the leading coefficient is 1. Therefore, the possible rational roots are the factors of 200, which include $\pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 25, \pm 40, \pm 50, \pm 100, \pm 200$ .

We can now use synthetic division to test these potential roots. Let's start with -2:

-2 | 1  6  33  150  200
   |   -2  -8  -50 -200
   ----------------------
     1  4  25  100   0

The remainder is 0, so -2 is a root, and $(x + 2)$ is a factor. The quotient is $x^3 + 4x^2 + 25x + 100$ .

Now, let's apply synthetic division again to the quotient $x^3 + 4x^2 + 25x + 100$ . Let's try -4:

-4 | 1  4  25  100
   |   -4   0 -100
   ----------------
     1  0  25   0

The remainder is 0, so -4 is a root, and $(x + 4)$ is a factor. The new quotient is $x^2 + 25$ .

The quadratic $x^2 + 25$ cannot be factored further using real numbers because it has no real roots. It is a sum of squares, which is irreducible over the real numbers. However, it can be factored using complex numbers as $(x + 5i)(x - 5i)$ , but we are looking for factorization with real coefficients in this case.

Therefore, the factored form of the polynomial $x^4 + 6x^3 + 33x^2 + 150x + 200$ is $(x + 2)(x + 4)(x^2 + 25)$ .

Conclusion

Synthetic division is a powerful tool for factoring polynomials, particularly those of higher degrees. It provides an efficient method for dividing a polynomial by a linear factor, identifying roots, and ultimately expressing the polynomial as a product of simpler factors. By understanding the Remainder Theorem and the Factor Theorem, we can effectively use synthetic division to factor polynomials and solve related problems.

In this guide, we have explored the intricacies of synthetic division, from its basic steps to its application in factoring a specific polynomial. We have seen how synthetic division, combined with the Rational Root Theorem, allows us to systematically find roots and factors. With a solid grasp of these concepts, you are well-equipped to tackle a wide range of polynomial factorization problems.

In summary, the correct answer to the question, "Using synthetic division, what is the factored form of this polynomial? $x^4 + 6x^3 + 33x^2 + 150x + 200$ ", is:

B. $(x + 2)(x + 4)(x^2 + 25)$