Question

Find the unknown factor. $$25 k^{7}-30 k^{5}-20 k^{2}=5 k^{2}(?)$$

Short answer

Answer: The unknown factor is \(5 k^{5} - 6 k^{3} - 4\).

Step-by-step solution

Identify Common Factors

In this problem, we have the expression \(25 k^{7}-30 k^{5}-20 k^{2}\). The common factors in all of these terms are \(5\) and \(k^{2}\).

Factor out Common Factors

Now, we will factor out the common factors that we identified in Step 1. We will rewrite the expression as \(5 k^{2}(a k^{5} - b k^{3} - c)\).

Determine Individual Factors

In order to determine the individual factors \(a\), \(b\), and \(c\), we will rewrite the original expression by factoring out the common factors:$$5 k^{2}(a k^{5} - b k^{3} - c) = 25 k^{7} - 30 k^{5} - 20 k^{2}$$Now, we will find the values of \(a\), \(b\), and \(c\) by comparing the terms on both sides of the equation. $$a k^{5} \cdot 5 k^{2} = 25 k^{7}$$$$b k^{3} \cdot 5 k^{2} = 30 k^{5}$$$$c \cdot 5 k^{2} = 20 k^{2}$$By comparing the coefficients and exponents, we find the factors to be: $$a = 5$$$$b = 6$$$$c = 4$$

Write the Final Expression

Now that we have determined the values for \(a\), \(b\), and \(c\), we can write the final expression for the unknown factor: $$5 k^{2}(5 k^{5} - 6 k^{3} - 4)$$So, the unknown factor is \(5 k^{5} - 6 k^{3} - 4\).

Key concepts

Common Factors

The concept of common factors plays a crucial role in simplifying algebraic expressions through the process of factoring. A common factor is a number, variable, or combination of both that is present in each term of a polynomial. For example, in the expression 25k^{7} - 30k^{5} - 20k^{2}, both 5 and k^{2} occur in every term, making them common factors of the polynomial.

Finding common factors is often the first step when factoring polynomials, as it allows us to rewrite the expression in a more compact form. Doing this can simplify calculations, reveal underlying patterns, and help in solving equations. It's kind of like breaking down a complex machine into its basic components to understand how it works more easily.

• Identify elements that every term of the polynomial shares.
• Use these elements to rewrite the polynomial as a product of the common factor and another polynomial.

Factorization

The factorization of polynomials is akin to deconstructing a complex structure into simpler building blocks. It is an essential skill in algebra that involves expressing a polynomial as a product of its factors, which could be numbers, variables, or other polynomials.

Factorization not only simplifies algebraic expressions but also is vital in solving equations and understanding the behavior of graphs. A common method of factorization is to start with finding common factors and then proceed to use various techniques such as grouping, the quadratic formula, or special products. It's like solving a puzzle: starting with the obvious pieces and then working out where the more intricate ones fit in.

Algebraic Expressions

Understanding algebraic expressions is a cornerstone of algebra. They are combinations of numbers, variables (letters that represent numbers), and arithmetic operations (like addition and multiplication). Polynomials are a type of algebraic expression that have variables raised to whole number exponents, and possibly several terms added or subtracted together.

An expression like 25k^{7} - 30k^{5} - 20k^{2} is an example of a polynomial. Each separate term, such as 25k^{7}, is like a distinct 'ingredient' in the recipe that makes up the whole expression. Manipulating these expressions using operations like factorization allows us to solve complex equations and problems efficiently.

Exponents

Exponents are shorthand for repeated multiplication. When we work with algebraic expressions, exponents tell us how many times to multiply the base (the number or variable) by itself. For example, k^{2} means k * k, and k^{7} would mean k * k * k * k * k * k * k.

Exponents are crucial when factoring polynomials, as we often need to manipulate expressions with variables raised to different powers. When factoring, it's important to be familiar with exponent rules, such as the product of powers rule, which states that when we multiply two powers with the same base, we add the exponents. This rule is clearly at play when the expression 25k^{7} is factored into 5k^{2} * 5k^{5}, where exponents 2 and 5 add up to 7.