Problem 36 Factor each polynomial. $$ 6... [FREE SOLUTION]

Chapter 5: Problem 36

Factor each polynomial. $$ 6 x^{2}-8 $$

Short Answer

Expert verified

The polynomial $6x^2 - 8$ factors to $2(3x^2 - 4)$.

Step by step solution

Identify Common Factors

First, observe the terms $6x^2$ and $-8$. Both terms have a common factor of 2. Recognizing common factors will help simplify the expression. Thus, we can factor out 2 from the polynomial, which results in $2(3x^2 - 4)$.

Check for Further Factoring

Next, look at the expression $3x^2 - 4$ inside the parentheses. Check if it can be factored further, such as by recognizing any perfect squares or differences of squares. In this case, $3x^2 - 4$ cannot be factored further over the integers, as it's not a difference of squares and does not share further common factors.

Write the Final Factored Form

Since $3x^2 - 4$ cannot be factored further, the expression $2(3x^2 - 4)$ is the fully factored form of the original polynomial $6x^2 - 8$.

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Key Concepts

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

Common Factors in Polynomial Factoring

Finding common factors in polynomials is often the first step in simplifying expressions. A common factor is a number or term that divides two or more terms without leaving a remainder. In the polynomial expression $6x^2 - 8$, we look at each term: 6x虏 and 8. Each of these numbers shares a numerical factor beyond 1.

Identify numbers that divide both coefficients evenly. In this case, both terms are divisible by 2.
Factor out this common factor from the polynomial. Here, factoring out 2 gives us $2(3x^2 - 4)$.

By doing this, you simplify the polynomial and make it easier to tackle more complex factoring scenarios. Always start with this step, as it's a straightforward way to reduce complexity.

Recognizing Differences of Squares

The difference of squares is a special case in factoring where an expression is written as two squares subtracted from each other. The general form looks like $a^2 - b^2$ and can be factored into $(a-b)(a+b)$. Recognizing this form helps simplify expressions effectively.

Check if both terms are perfect squares.
Ensure the expression involves subtraction.

In the polynomial $3x^2 - 4$, had it fit the difference of squares pattern, we could have factored it further. However, $3x^2 - 4$ isn't a difference of squares, because 3 isn't a perfect square and the expression cannot be broken down to meet the criteria. Understanding this concept is crucial to efficiently factor expressions when applicable.

Understanding Perfect Squares in Polynomials

Perfect squares, when it comes to polynomials, refer to expressions that are squares of binomials. For example, a perfect square trinomial like $(x + y)^2$ expands to $x^2 + 2xy + y^2$. This structure doesn't apply to difference operations but reflects important symmetry and balance.

Recognize expressions of the form $a^2 + 2ab + b^2$.
Factor them back into $(a+b)^2$ or $(a-b)^2$ when applicable.

In the expression $3x^2 - 4$, it's not a perfect square because it doesn't fit the classic binomial square patterns. Thus, it was already simplified by factoring out common factors only. Mastery over this concept aids in recognizing when not to overly complicate straightforward factoring.

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91影视

Factor each polynomial. $$ 6 x^{2}-8 $$

Short Answer

Step by step solution

Identify Common Factors

Check for Further Factoring

Write the Final Factored Form

Key Concepts

Common Factors in Polynomial Factoring

Recognizing Differences of Squares

Understanding Perfect Squares in Polynomials

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