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Chapter 6: Problem 117

Factor. $$ 2 x^{3} z-4 x^{2} z+32 x z-64 z $$

Short Answer

Expert verified
The expression factors to \(2z(x - 2)(x^2 + 16)\).

Step by step solution

01

Factor out the Greatest Common Factor (GCF)

Look at all the terms in the expression: \(2x^3z\), \(-4x^2z\), \(32xz\), and \(-64z\). Each term contains a \(z\), so the GCF of all terms is \(2z\). Let's factor \(2z\) out of the expression:\[2z(x^3 - 2x^2 + 16x - 32).\]
02

Simplify the Expression Inside the Parentheses

Focus on the expression inside the parentheses: \(x^3 - 2x^2 + 16x - 32\). Arrange and group the terms: \((x^3 - 2x^2) + (16x - 32)\).
03

Factor by Grouping

Now, let's factor each group separately.- In \(x^3 - 2x^2\), factor out \(x^2\): \[x^2(x - 2) \]- In \(16x - 32\), factor out \(16\): \[16(x - 2) \]So we have:\[(x^2)(x - 2) + 16(x - 2)\]
04

Factor out the Common Binomial Factor

Notice that \((x - 2)\) is a common factor in both terms. Factor \((x - 2)\) from both:\[(x - 2)(x^2 + 16)\]
05

Combine All Factored Terms

Now include the GCF from Step 1 with the factored expression:\[2z(x - 2)(x^2 + 16).\]

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Key Concepts

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

Greatest Common Factor
Factoring a polynomial often starts with identifying the Greatest Common Factor (GCF). The GCF is the largest factor shared by all terms in a polynomial. To find it, examine all coefficients and variable parts of each term.
Most polynomial problems provide terms having common numerical coefficients or variables.
  • For instance, look at the expression \(2x^3z - 4x^2z + 32xz - 64z\). Here, each term contains a \(z\). We also identify the numerical GCF. Comparing coefficients \(2, -4, 32,\) and \(-64\), the number \(2\) is identified as the GCF of the coefficients.
  • Combine these observations to factor \(2z\) out from each term, simplifying the polynomial to \(2z(x^3 - 2x^2 + 16x - 32)\).
Recognizing the GCF is crucial, as it simplifies the polynomial and lays groundwork for further factoring methods.
Factor by Grouping
Factoring by grouping is a helpful tactic when a polynomial is not easily factorable using the GCF alone. This method involves rearranging the polynomial into groups with a common factor.
Applying this to the expression \(x^3 - 2x^2 + 16x - 32\), grouping strategies become key to simplifying further.
  • First, rearrange into two pairs: \((x^3 - 2x^2)\) and \((16x - 32)\).
  • For \(x^3 - 2x^2\), factor out \(x^2\) to get \(x^2(x - 2)\).
  • Similarly, in \(16x - 32\), use the common factor \(16\): \(16(x - 2)\).
Through this process, the expression simplifies into \[x^2(x - 2) + 16(x - 2)\].
This grouping technique reveals underlying common factors, essential for simplifying complex polynomials.
Binomial Factor
After grouping, often a binomial factor emerges. By recognizing this shared binomial, the polynomial can be further reduced by factoring it out.
In our expression, the polynomial \(x^2(x - 2) + 16(x - 2)\) presents \((x - 2)\) as a common binomial factor.
  • Focus on the shared binomial: both terms include \((x - 2)\).
  • Factor out \((x - 2)\), simplifying further into \[(x - 2)(x^2 + 16)\].
The binomial factor step efficiently condenses expressions, revealing a concise product form.
To wrap it all up, reintroduce any previously factored constants or terms from initial steps. This process culminates as \[2z(x - 2)(x^2 + 16)\].
Understanding binomial factors is critical to factor reduction and simplification within polynomials.

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