Factor – Definition with Examples

Table of Contents

    Factors are one of the foundational concepts in math, particularly in algebra and arithmetic. In this article, you’ll learn what is a factor, how to factor equations, and how to factor a polynomial.

    What Are Factors?

    So, what is a factor in math? Factors are like the building blocks of numbers. They are little puzzle pieces that fit snugly together to create a larger mathematical picture. You can express factor definition as the integers that can be divided evenly into a given number without leaving any remainder. Let’s factor 75 for example. The factors of 75 are the numbers that can divide 75 without a remainder.

    The factors of 75 are 1, 3, 5, 15, 25, and 75. These numbers can be multiplied in pairs to give the product of 75. For example:

    1 × 75 = 75

    3 × 25 = 75

    5 × 15 = 75

    Thus, these pairs of numbers are the factors of 75.

    Factor Theorem

    The Factor Theorem is a powerful tool in algebra that helps us determine if a given binomial is a factor of a polynomial. Factor theorem states that if we substitute a potential factor into the polynomial and the result is equal to zero, then that binomial is indeed a factor. Let’s see an example:

    Example: Determine if (x – 2) is a factor of the polynomial 3x^2 – 8x + 4.

    Solution: Substitute (x – 2) into the polynomial: 3(2)^2 – 8(2) + 4 = 0. Since the result is zero, (x – 2) is a factor of the polynomial.

    Properties of Factors

    Factors possess a fascinating array of properties or characteristics that make them intriguing. Let’s delve into some of these captivating traits:

    • Multiplicity: A factor can appear multiple times in a given number. For example, the number 12 has 1, 2, 3, 4, 6, and 12 as its factors, which shows the concept of multiplicity.
    • Pairing: Factors come in pairs, with each pair multiplying to form the given number. You can think of it as some type of mathematical duet, with each factor harmonizing with its partner to create a melodious result.
    • Divisibility: Factors are divisible by the number they belong to. If a number is divisible by another number without leaving a remainder, then it is a factor of that number. This property is a practical way to identify factors.

    How to Find Factors?

    Now that we have a grasp of what factors are, let’s explore some practical methods for finding factors. They include: 

    • Prime Factorization: This method involves breaking down a number into its prime factors, which are the building blocks of that number. When we identify prime factors and their respective powers, we can determine all the factors.
    • Divisibility Tests: Divisibility rules are handy tools for finding factors. When we apply these rules, including divisibility by 2, 3, 5, and so on, we can quickly identify potential factors.
    • Listing and Testing: Another approach involves systematically listing and testing numbers to determine if they divide evenly into the given number. Starting from 1 and progressing upwards, we check if each number is a factor.

    How to Find Factors of a Number?

    Here’s how to find the factors of a number:

    • Start at the Beginning: Start by identifying the number itself. Every number is divisible by 1 and itself, so these two values are always factors.
    • Be Systematic: Proceed systematically by testing numbers in ascending order. Start with 2 and move on to 3, 4, 5, and so forth. Remember, we can skip even numbers (except 2) since they cannot divide an odd number evenly.
    • The Art of Division: Divide the number by each potential factor. If the division yields no remainder, voila! You’ve found a factor.
    • Capture the Pair: Don’t forget to capture the paired factor as well. For instance, if 2 is a factor, remember that the number itself is also a factor.

    Solved Problems on Factors

    • Find the factors of 42.

    Solution: The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

    • Determine if 15 is a factor of 105.

    Solution: Yes, 15 is a factor of 105 because 105 ÷ 15 = 7 with no remainder.

    • Find the common factors of 24 and 36.

    Solution: The common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12.

    Factoring Polynomials

    In this section, we’ll learn about factor polynomials. A polynomial is an expression consisting of variables, coefficients, and mathematical operations. To factorize a polynomial, we aim to break it down into its simplest form, which involves finding its factors. Let’s explore an example on how to factor polynomials.

    Example: Factor the polynomial x^2 + 5x + 6.

    Solution: To factorize the polynomial, we need to find two binomials that, when multiplied, will give us the original polynomial. In this case, we can factorize it as (x + 2)(x + 3).

    How to Factor Quadratic Equations

    Quadratic equations are a special type of a polynomial with the highest degree of 2. Factoring quadratic equations can help us find their roots, which are the values that make the equation equal to zero. 

    To factor a quadratic equation, we look for two binomials that, when multiplied together, give us the original quadratic expression. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.

    Follow these steps to learn how to factor quadratic equations:

    • Check if the quadratic expression can be factored by finding common factors. If there are common factors, factor them out first.
    • Write down two sets of parentheses.
    • Identify two numbers that multiply to give the product of the coefficient of x^2 term (a) and the constant term (c), and add up to give the coefficient of the x term (b).
    • Use these two numbers to fill in the parentheses, placing the appropriate variables (usually x) in each term.
    • Simplify the expression by multiplying the terms inside the parentheses.
    • Set each factor equal to zero and solve for x to find the solutions of the quadratic equation.

    Example: Factor the quadratic equation x^2 – 4x – 5 = 0.

    Solution: 

    To factor the quadratic equation x^2 – 4x – 5 = 0, we need to find two binomials that, when multiplied together, give us the original quadratic expression. 

    Steps: 

    • Write down two sets of parentheses: ( )( ).
    • We need to find two numbers that multiply to give -5 (the product of the coefficient of x^2 term and the constant term) and add up to give -4 (the coefficient of the x term). These numbers are -5 and +1.
    • Fill in the parentheses with the appropriate signs and variables: (x – 5)(x + 1).
    • Multiply the terms inside the parentheses: x^2 – 5x + x – 5.
    • Simplify the expression: x^2 – 4x – 5.

    So, the factored form of the quadratic equation x^2 – 4x – 5 = 0 is (x – 5)(x + 1) = 0.

    To find the solutions, we can set each factor equal to zero:

    x – 5 = 0 or x + 1 = 0.

    Solving these equations gives us:

    x = 5 or x = -1.

    Therefore, the solutions to the quadratic equation x^2 – 4x – 5 = 0 are x = 5 and x = -1.

    Scale Factor Calculator

    The scale factor is a ratio that compares the dimensions of two similar objects or figures. A scale factor calculator can be a useful tool for determining the ratio of corresponding sides or dimensions between two similar objects or shapes. It helps in understanding the relationship between the original object and the scaled version.

    To use a scale factor calculator, you need to input the lengths of the corresponding sides in the original and scaled object. The calculator then determines the ratio and provides the scale factor. 

    Factors of Perfect Numbers

    A perfect number is a positive integer that is equal to the sum of its proper factors. The factors of a number exclude the number itself. For example, the number 6 is a perfect number because its factors (excluding 6) are 1, 2, and 3, and their sum is indeed 6.

    Factors Examples

    To solidify our understanding, let’s explore a few more examples:

    Example 1: Find the factors of 16.

    Solution: The factors of 16 are 1, 2, 4, 8, and 16.

    Example 2: Determine if 5 is a factor of 35.

    Solution: Yes, 5 is a factor of 35 because 35 ÷ 5 = 7 with no remainder.

    Example 3: Find the factors of 100.

    Solution: The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.

    Example 4: Determine if 9 is a factor of 27.

    Solution: Yes, 9 is a factor of 27 because 27 ÷ 9 = 3 with no remainder.

    Example 5: Find the factors of 81.

    Solution: The factors of 81 are 1, 3, 9, 27, and 81.

    Frequently Asked Questions

    What are the factors of 12?

    The factors of 12 are 1, 2, 3, 4, 6, and 12.

    How do you explain factors?

    Factors are numbers that can be multiplied together to obtain a given number. They divide the number without leaving a remainder.

    What are the factors of 27?

    The factors of 27 are 1, 3, 9, and 27.

    What are the factors of 42?

    The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

    What are the factors of 15?

    The factors of 15 are 1, 3, 5, and 15.

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