Problem 131 Factor each polynomial completel... [FREE SOLUTION]

Chapter 5: Problem 131

Factor each polynomial completely. $$4 b^{2}-4 a b+a^{2}$$

Short Answer

Expert verified

$ 4b^2 - 4ab + a^2 = (2b - a)^2 $

Step by step solution

- Identify a common factor and rewrite the polynomial

First, identify if there is any common factor in all the terms of the polynomial. However, in this case, there is no common numerical factor. So, proceed to rewriting the polynomial to see if it fits a recognizable factoring pattern.

- Recognize the form of the polynomial

Observe that the given polynomial resembles the format of a perfect square trinomial. Recall that $ A^2 - 2AB + B^2 = (A - B)^2 $.

- Rewrite the polynomial in a perfect square trinomial format

Rewrite the polynomial $ 4b^2 - 4ab + a^2 $ as $ (2b)^2 - 2(2b)(a) + a^2 $.

- Factor the trinomial

Apply the perfect square trinomial pattern: $ (A - B)^2 $. In this case, it becomes $ (2b - a)^2 $.

- Write the final factored form

Thus, the polynomial $ 4b^2 - 4ab + a^2 $ can be factored as $ (2b - a)^2 $.

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Key Concepts

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

Perfect Square Trinomial

A perfect square trinomial is a special kind of polynomial. It is formed when a binomial is squared. In other words, it looks like $A^2 - 2AB + B^2$ or $A^2 + 2AB + B^2$. These trinomials can always be factored into the square of a binomial. For example, $4b^2 - 4ab + a^2$ can be recognized as a perfect square trinomial.

Polynomial Factoring

Factoring polynomials means rewriting them as a product of simpler polynomials. This helps in solving equations and simplifying expressions. To factor effectively, one must recognize patterns and apply appropriate factoring techniques. For example, the polynomial $4b^2 - 4ab + a^2$ can be rewritten as $(2b - a)^2$. Identifying this pattern allows for a straightforward factorization.

Factorization Patterns

Factorization patterns are useful tools that make factoring polynomials easier. These patterns include:

Common factor: Finding a common number or variable in all terms.
Difference of squares: For example, $A^2 - B^2 = (A+B)(A-B)$.
Perfect square trinomials: As seen earlier, $A^2 - 2AB + B^2 = (A-B)^2$ and $A^2 + 2AB + B^2 = (A+B)^2$.
Sum and difference of cubes.

By recognizing these patterns, the factorization process becomes much simpler and more efficient.

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91影视

Factor each polynomial completely. $$4 b^{2}-4 a b+a^{2}$$

Short Answer

Step by step solution

- Identify a common factor and rewrite the polynomial

- Recognize the form of the polynomial

- Rewrite the polynomial in a perfect square trinomial format

- Factor the trinomial

- Write the final factored form

Key Concepts

Perfect Square Trinomial

Polynomial Factoring

Factorization Patterns

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