Problem 44 Factor completely. If a polynomi... [FREE SOLUTION]

Chapter 6: Problem 44

Factor completely. If a polynomial is prime, state this. $$ 20 a^{2}-23 a x+6 x^{2} $$

Short Answer

Expert verified

(t + 1 - 9r)(t + 1 + 9r)

Step by step solution

Identify the polynomial structure

The given polynomial is $ t^{2} + 2t + 1 - 81r^{2} $. Notice that $ t^{2} + 2t + 1 $ is a perfect square trinomial, and $ 81r^{2} $ is a perfect square.

Rewrite the polynomial using perfect squares

Rewrite the polynomial as $ (t^2 + 2t + 1) - (81r^2) $. Recognize that $ t^2 + 2t + 1 $ is $(t + 1)^2$ and $ 81r^2 $ is $ (9r)^2 $. So it becomes $ (t + 1)^2 - (9r)^2 $.

Apply the difference of squares formula

Use the difference of squares formula: $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = t + 1 $ and $ b = 9r $. Therefore, $ (t + 1)^2 - (9r)^2 = (t + 1 - 9r)(t + 1 + 9r) $.

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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 quadratic expression that can be written as the square of a binomial. For example, the polynomial $ t^2 + 2t + 1 $ is a perfect square trinomial because it can be expressed as $(t + 1)^2$. To identify a perfect square trinomial, look for a quadratic term $t^2$, a linear term $2t$, and a constant term $1$. Here, the constant term should be the square of half the coefficient of the linear term. This type of trinomial forms the basis of many factoring problems and helps in converting complex expressions into simpler forms.

Difference of Squares

The difference of squares is a special algebraic identity that states $ a^2 - b^2 = (a - b)(a + b) $. This identity is useful for factoring expressions that can be written as the difference between two perfect squares. In the provided exercise, the polynomial $ (t+1)^2 - (9r)^2 $ is structured as a difference of squares. Here, we can set $ a = t+1 $ and $ b = 9r $. This allows the polynomial to be factored into $ (t+1 - 9r)(t+1 + 9r) $. Recognizing patterns like these simplifies the factoring process and is a valuable skill in algebra.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They are tools that simplify algebraic expressions and are particularly useful in factoring and expanding polynomials. Key identities include the perfect square trinomial $ (a + b)^2 = a^2 + 2ab + b^2 $ and the difference of squares $ a^2 - b^2 = (a - b)(a + b) $. In our exercise, applying these identities step-by-step allows us to break down a seemingly complex polynomial into simpler factors. Mastery of these identities is crucial for tackling a variety of algebraic problems and achieving proficiency in algebra.

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91影视

Factor completely. If a polynomial is prime, state this. $$ 20 a^{2}-23 a x+6 x^{2} $$

Short Answer

Step by step solution

Identify the polynomial structure

Rewrite the polynomial using perfect squares

Apply the difference of squares formula

Key Concepts

Perfect Square Trinomial

Difference of Squares

Algebraic Identities

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