Problem 77 Factor. $$15 x^{4} y^{2}-13 x^... [FREE SOLUTION]

Chapter 10: Problem 77

Factor. $$15 x^{4} y^{2}-13 x^{3} y^{3}-20 x^{2} y^{4}$$

Short Answer

Expert verified

The simplified and factored expression is $x^{2}y^{2}(15 x^{2} - 13 x y - 20)$

Step by step solution

Identify the GCF

The GCF is determined by finding out the common factors between the coefficients (15,-13,-20) and the lowest power of the variables x and y in the polynomial. The common factor of the coefficients is only 1. For the variable 'x', the lowest power is $x^2$, and for the variable 'y', the lowest power is $y^2$. Therefore, the GCF is $x^2 y^2$.

Pair the GCF with each term

Take this GCF and divide each term by it. The resulting factor pairs are: $15x^2$ , $-13xy$ and -20.

Write the final factored expression

Now, write down the answer by multiplying the GCF with the paired factors placed inside the parentheses: $x^{2}y^{2}(15 x^{2} - 13 x y - 20)$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor

In the realm of algebra, one of the fundamental steps in simplifying polynomials is identifying the Greatest Common Factor, often abbreviated as GCF. The GCF of a polynomial is the largest expression that evenly divides all of the terms in the polynomial. To find the GCF, you should:

Look at the coefficients of each term. For example, in the expression provided, the coefficients are 15, -13, and -20. The greatest common factor from these numbers is 1, as they have no other common factors besides 1.
Identify the lowest power of each variable present in all terms. In this given exercise, the terms contain the variables 'x' and 'y'. The lowest exponents are $x^2$ and $y^2$, making $x^2y^2$ the GCF.

Understanding how to find the GCF is crucial because it simplifies the polynomial expression, making it easier to work with in subsequent steps.

Polynomial Expressions

A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Simple polynomials include terms like $x^{2}$, $xy$, and constants, while more complex polynomials may involve higher degrees and multiple variables. There are several key features of polynomial expressions:

Terms: Each individual part of a polynomial separated by a plus or minus sign is called a term. For example, in $15x^{4}y^{2} - 13x^{3}y^{3} - 20x^{2}y^{4}$, there are three terms.
Degree: The degree of a polynomial is the highest sum of the exponents in a term. In our example, the term $15x^{4}y^{2}$ has a degree of 6 (since $4 + 2 = 6$).

Understanding polynomial expressions is essential as it sets the foundation for more advanced algebraic techniques, such as factoring and solving equations.

Algebraic Factoring

Algebraic factoring is a method used to write a polynomial as a product of its factors. This process simplifies expressions and helps in solving polynomial equations. In our exercise, the factorization involves several steps:

Identify the GCF: As discussed earlier, compute the GCF of the polynomial. Here, it is $x^2y^2$.
Divide Each Term by the GCF: Once found, divide each term of the polynomial by the GCF. For example, $15x^{4}y^{2}$ divided by $x^2y^2$ is $15x^2$.
Write the Factored Expression: Combine the GCF with the new terms inside parentheses to get the factored polynomial. In this case, the expression becomes $x^2y^2(15x^2 - 13xy - 20)$.

Algebraic factoring is a powerful tool in algebra that allows complex expressions to be handled more easily, and it proves extremely useful in various applications like integration, simplification, and solving equations.

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91影视

Factor. $$15 x^{4} y^{2}-13 x^{3} y^{3}-20 x^{2} y^{4}$$

Short Answer

Step by step solution

Identify the GCF

Pair the GCF with each term

Write the final factored expression

Key Concepts

Greatest Common Factor

Polynomial Expressions

Algebraic Factoring

One App. One Place for Learning.

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