How to Factor Polynomials
Understanding the Basics of Factoring Polynomials
Before diving into more complex techniques for factoring polynomials, it’s important to understand the basics of what factoring means. Factoring is the process of finding two or more expressions that, when multiplied together, result in a given polynomial.
For example, consider the polynomial x^2 + 5x + 6. We can factor this polynomial by finding two expressions that, when multiplied together, result in x^2 + 5x + 6. One possible factorization is (x + 3)(x + 2), since (x + 3)(x + 2) = x^2 + 5x + 6.
Factoring can be useful for simplifying expressions, solving equations, and finding roots of polynomials. It can also help us identify patterns and relationships between different polynomials. By understanding the basics of factoring, we can build a solid foundation for more advanced techniques.
Factoring Common Polynomial Types
Certain types of polynomials have factorizations that are commonly used and should be memorized. These include:
Monomials: A monomial is a polynomial with only one term. Monomials can be factored by extracting their greatest common factor. For example, the monomial 3x^2 can be factored into 3(x^2).
Quadratics: A quadratic is a polynomial of degree two. Quadratics can be factored using techniques such as factoring by grouping, completing the square, or using the quadratic formula. For example, the quadratic x^2 + 5x + 6 can be factored into (x + 3)(x + 2).
Cubics: A cubic is a polynomial of degree three. Cubics can be factored using techniques such as grouping or using the rational roots theorem. For example, the cubic x^3 + 3x^2 + 2x can be factored into x(x + 1)(x + 2).
Difference of Squares: A difference of squares is a polynomial of the form a^2 – b^2. A difference of squares can be factored into (a + b)(a – b). For example, the polynomial x^2 – 9 can be factored into (x + 3)(x – 3).
By memorizing these common factorizations, we can save time and simplify the factoring process.
Factoring by Grouping and Completing the Square
In addition to the common polynomial types, there are other techniques for factoring that can be useful in certain situations. Two of these techniques are factoring by grouping and completing the square.
Factoring by grouping is a technique used when a polynomial has four terms. To factor by grouping, we group the terms into two pairs and look for a common factor in each pair. We then factor out that common factor and see if we can factor the resulting expression further. For example, consider the polynomial x^3 – x^2 + x – 1. We can group the first two terms and the last two terms and factor out x^2 and 1, respectively. This gives us (x^2 – 1)(x – 1). We can further factor the first term using the difference of squares rule, giving us (x + 1)(x – 1)(x – 1).
Completing the square is a technique used to factor quadratics of the form ax^2 + bx + c. To complete the square, we add and subtract a term (b/2a)^2 to the quadratic. This creates a perfect square trinomial that can be factored using the square root property. For example, consider the quadratic x^2 + 6x + 7. We can complete the square by adding and subtracting (6/2)^2 = 9. This gives us x^2 + 6x + 9 – 9 + 7, which can be factored into (x + 3)^2 – 2.
By using these techniques, we can factor more complex polynomials and expand our factoring toolkit.
Using the Rational Roots Theorem to Factor Polynomials
The rational roots theorem is a useful tool for factoring polynomials with integer coefficients. This theorem states that if a polynomial with integer coefficients has a rational root, then that root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
To use the rational roots theorem, we start by listing all the factors of the constant term and all the factors of the leading coefficient. We then form all possible fractions using a factor from the constant term and a factor from the leading coefficient. We test each of these possible roots by plugging them into the polynomial and seeing if the result is zero. If we find a rational root, we can use synthetic division or long division to factor the polynomial.
For example, consider the polynomial x^3 – 3x^2 – 4x + 12. The factors of the constant term 12 are 1, 2, 3, 4, 6, and 12. The factors of the leading coefficient 1 are 1 and -1. This gives us the possible roots 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, and -12. By testing these roots, we find that x = 3 is a root. Using synthetic division, we can then factor the polynomial into (x – 3)(x^2 – 4).
The rational roots theorem can be a powerful tool for factoring polynomials, but it is limited to polynomials with integer coefficients and rational roots.
Strategies for Factoring Complex Polynomials
For polynomials that are not easily factored using the techniques described earlier, there are several strategies we can use to factor them. Here are some examples:
Factoring by Substitution: This technique involves substituting a variable expression for a specific term in the polynomial. For example, consider the polynomial x^3 + 4x^2 + 4x + 1. We can substitute y = x + 1 to get the polynomial in terms of y: (y – 1)^3. We can then factor the resulting expression as a cube of a binomial: (y – 1)^3 = (y – 1)(y – 1)(y – 1) = (y – 1)^3 = (x + 1 – 1)^3 = x^3.
Factoring by Long Division: For polynomials that cannot be factored using other techniques, long division can be used to factor the polynomial into simpler polynomials. For example, consider the polynomial x^4 – 3x^3 – 4x^2 + 12x + 4. By using long division, we can factor this polynomial into (x – 2)(x + 1)(x^2 – 2x – 2).
Factoring Using Trigonometric Functions: Certain polynomials can be factored using trigonometric functions, such as sin(x) and cos(x). For example, consider the polynomial x^4 – 4x^2 + 4. This can be factored into (x^2 – 2)(x^2 – 2sin^2(x)).
Factoring Using Special Identities: Certain special identities, such as the difference of cubes or the sum and difference of cubes, can be used to factor polynomials. For example, consider the polynomial x^3 + 8. This can be factored into (x + 2)(x^2 – 2x + 4) using the sum of cubes identity.
By using these strategies, we can factor even the most complex polynomials and solve difficult problems in mathematics and science.