Problem 76 Factor by grouping. \(8 s^{2}-... [FREE SOLUTION]

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Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 6: Problem 76

Factor by grouping. $8 s^{2}-4 s t+6 s y-3 y t$

Short Answer

Expert verified

(2s - t)(4s + 3y)

Step by step solution

- Group the terms

Group the four terms in pairs that can be factored separately. For the expression, group it as follows: (8s^2 - 4st) + (6sy - 3yt).

- Factor out the Greatest Common Factor (GCF) from each group

Factor out the GCF from each of the grouped pairs. The first group (8s^2 - 4st) has a GCF of 4s, and the second group (6sy - 3yt) has a GCF of 3y: 4s(2s - t) + 3y(2s - t).

- Factor out the common binomial

Now, notice that each term has a common binomial factor of (2s - t). Factor this out from the expression: (2s - t)(4s + 3y).

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Key Concepts

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

algebraic expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. For example, in the expression $8s^{2}-4st+6sy-3yt$, the numbers (like 8, 4, 6, and 3) are constants, and the letters (s, t, y) are variables.
Operations can include addition, subtraction, multiplication, and so on. When working with algebraic expressions, we often aim to simplify them or solve them if they are equations.
Simplifying can involve several techniques, such as combining like terms and factoring.

greatest common factor

The greatest common factor, or GCF, is the largest number that divides two or more terms without leaving a remainder.
When factoring by grouping, we look for the GCF in pairs of terms. For example, in the expression $8s^{2}-4st+6sy-3yt$, we group the terms into $ (8s^2 - 4st) $ and $ (6sy - 3yt) $.
In the first group $8s^2 - 4st$, the GCF is $4s$, and in the second group $6sy - 3yt$, the GCF is $3y$.
Factoring out the GCF makes the expression easier to work with by reducing it into simpler terms.

factoring

Factoring is the process of breaking down an expression into simpler components that, when multiplied together, give the original expression.
Factoring by grouping is a common method, especially useful for four-term polynomials. Start by grouping the terms: $ (8s^2 - 4st) + (6sy - 3yt) $.
Then, factor out the GCF from each group: $4s(2s - t) + 3y(2s - t) $.
Finally, factor out the common binomial factor $(2s - t)$: $(2s - t)(4s + 3y)$.
This final expression is fully factored and much simpler to work with.

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91影视

Factor by grouping. \(8 s^{2}-4 s t+6 s y-3 y t\)

Short Answer

Step by step solution

- Group the terms

- Factor out the Greatest Common Factor (GCF) from each group

- Factor out the common binomial

Key Concepts

algebraic expressions

greatest common factor

factoring

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