Problem 60 Factor completely. Identify any ... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 6: Problem 60

Factor completely. Identify any prime polynomials. $$ 3 w^{2}-2700 $$

Short Answer

Expert verified

The completely factored form is $3(w - 30)(w + 30)$.

Step by step solution

Identify the GCD

Identify the greatest common divisor (GCD) of the coefficients in the polynomial. Here, the coefficients are 3 and 2700. The GCD is 3.

Factor out the GCD

Factor out the GCD from the polynomial: \[ 3w^{2} - 2700 = 3(w^{2} - 900) \]

Recognize a difference of squares

Notice that the expression inside the parentheses $w^2 - 900$ is a difference of squares. Recall that $a^2 - b^2 = (a - b)(a + b)$. Here, $w^2$ is $a^2$ and 900 is $b^2$, with $900 = 30^2$.

Apply the difference of squares formula

Rewrite the expression using the difference of squares formula: \[ w^{2} - 900 = (w - 30)(w + 30) \]

Substitute back into the factored form

Substitute the factored form back into the original equation: \[ 3(w^{2} - 900) = 3(w - 30)(w + 30) \]

Verify if the polynomial is prime

$3(w - 30)(w + 30)$ is the completely factored form. Since it is factored into linear binomials, the polynomial is not prime.

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Key Concepts

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

greatest common divisor

When factoring polynomials, the first step is often to identify the Greatest Common Divisor (GCD) of the coefficients. The GCD is the largest number that divides all of the coefficients without leaving a remainder. For example, in the polynomial $3w^2 - 2700$, the coefficients are 3 and 2700. The GCD is 3 because it is the largest number that evenly divides both 3 and 2700.

difference of squares

A difference of squares is a special polynomial form where two squared terms are subtracted. It has the general formula $a^2 - b^2 = (a - b)(a + b)$. In our example, after factoring out the GCD, we are left with $w^2 - 900$. Recognizing that 900 is a perfect square, specifically $30^2$, allows us to rewrite it as $w^2 - 30^2$. Using the difference of squares formula, we can factor it as $ (w - 30)(w + 30) $.

prime polynomial

A prime polynomial is a polynomial that cannot be factored into smaller polynomials with integer coefficients. In our example, we factored $3w^2 - 2700$ completely to get \ 3(w - 30)(w + 30) \. Since it can be broken down into linear binomials, it is not a prime polynomial. A polynomial is only prime if, after all attempts, you can't factor it any further using real numbers.

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91影视

Factor completely. Identify any prime polynomials. $$ 3 w^{2}-2700 $$

Short Answer

Step by step solution

Identify the GCD

Factor out the GCD

Recognize a difference of squares

Apply the difference of squares formula

Substitute back into the factored form

Verify if the polynomial is prime

Key Concepts

greatest common divisor

difference of squares

prime polynomial

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