Problem 231 Factor. \(75 u^{3}-30 u^{2} v+... [FREE SOLUTION]

Chapter 7: Problem 231

Factor. \(75 u^{3}-30 u^{2} v+3 u v^{2}\)

Short Answer

Expert verified

The factored form is \[3u(5u - v)^2\].

Step by step solution

Identify the Greatest Common Factor (GCF)

First, find the greatest common factor of the coefficients (75, 30, and 3). The GCF for these numbers is 3. Additionally, note that each term in the polynomial includes at least one factor of u. Therefore, the GCF of the entire expression is 3u.

Factor Out the GCF

Next, factor 3u out of each term in the polynomial. This gives: \[75u^{3} - 30u^{2}v + 3uv^{2} = 3u(25u^{2} - 10uv + v^{2}) \]

Factor the Quadratic Expression

Now, focus on factoring the quadratic expression inside the parentheses: \[25u^{2} - 10uv + v^{2}\]. This can be seen as a perfect square trinomial. The quadratic expression can be rewritten as: \[(5u - v)^2\].

Write the Final Factored Form

Combine the results from the previous steps to write the fully factored form of the polynomial: \[75u^{3} - 30u^{2}v + 3uv^{2} = 3u(5u - v)^{2}\]

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Key Concepts

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

Greatest Common Factor

When factoring polynomials, the first step is to identify the Greatest Common Factor (GCF). This is the largest number that each term in the polynomial can be divided by without leaving a remainder.

For our polynomial, \(75u^{3} - 30u^{2}v + 3uv^{2}\), we need to find the GCF of the coefficients 75, 30, and 3.

The GCF of these numbers is 3 because:

75 = 3 * 25
30 = 3 * 10
3 = 3 * 1

In addition, each term in the polynomial has at least one factor of 'u'. So, the GCF of the entire polynomial is 3u.

Factoring out the GCF simplifies the polynomial to:

75u^{3} - 30u^{2}v + 3uv^{2}= 3u(25u^{2} - 10uv + v^{2})

This is the expression we will work with in the next steps.

Quadratic Expression

A quadratic expression is a polynomial of degree 2, which means the highest power of the variable is squared. In this case, the quadratic expression inside the parentheses is: 25u^{2} - 10uv + v^{2}

To better understand how to factor this quadratic expression, look for patterns or special forms like perfect square trinomials.

Generally, a quadratic expression takes the form:

Ax^{2} + Bx + C

Where A, B, and C are constants.

Perfect Square Trinomial

A perfect square trinomial is a special kind of quadratic expression that can be written as the square of a binomial. For example, the expression 'a^{2} + 2ab + b^{2}' is a perfect square trinomial because it equals (a + b)^{2} In our example: 25u^{2} - 10uv + v^{2}

Notice the pattern that suggests a perfect square trinomial:

(5u)^{2} = 25u^{2}
2*(5u)(-v) = -10uv
(-v)^{2} = v^{2}

Putting these together, we can rewrite the quadratic expression as:

(5u - v)^{2} So, the fully factored form of the polynomial becomes:

3u(5u - v)^{2}

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91影视

Factor. \(75 u^{3}-30 u^{2} v+3 u v^{2}\)

Short Answer

Step by step solution

Identify the Greatest Common Factor (GCF)

Factor Out the GCF

Factor the Quadratic Expression

Write the Final Factored Form

Key Concepts

Greatest Common Factor

Quadratic Expression

Perfect Square Trinomial

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