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Chapter 5: Problem 57

Factor completely. \(2 x^{6}+8 x^{5}-42 x^{4}\)

Short Answer

Expert verified
2x^{4}(x + 7)(x - 3)

Step by step solution

01

Factor out the Greatest Common Factor (GCF)

Identify and factor out the GCF of the polynomial. The GCF of the terms in the polynomial is 2x^4. So, equation goes from: 2x^{6} + 8x^{5} - 42x^{4} to: 2x^{4}(x^{2} + 4x - 21)
02

Factor the quadratic expression

Focus on factoring the quadratic term inside the parenthesis: x^{2} + 4x - 21. To do this, find two numbers that multiply to -21 and add to 4. These numbers are 7 and -3. Therefore, we can write: x^{2} + 4x - 21 = (x + 7)(x - 3)
03

Combine the factors

Substitute the factors from step 2 back into the equation from step 1 to get the completely factored form: 2x^{4}(x + 7)(x - 3).This is the completely factored form of the given polynomial.

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) is the largest factor that divides each term of a polynomial. Identifying the GCF is the first step in factoring polynomials because it simplifies the expression for further factoring.
To find the GCF of a polynomial:
  • Identify the smallest power of each common variable in all terms.
  • Find the largest coefficient that can divide all the term coefficients evenly.
In the exercise, the GCF of \(2x^{6} + 8x^{5} - 42x^{4}\) is determined as 2x^{4}. This simplifies the polynomial to \(2x^{4}(x^{2} + 4x - 21)\), making it easier to factor the remaining quadratic expression.
Quadratic Expression
A quadratic expression is a polynomial of degree 2, generally taking the form \(ax^2 + bx + c\). In this example, inside the parenthesis, we have \(x^{2} + 4x - 21\).
To factor this quadratic expression, you need to find two numbers that both multiply to the constant term (-21) and add to the linear coefficient (4).
Here鈥檚 how it works:
  • List pairs of factors of -21: (1,-21), (-1,21), (3,-7), (-3,7).
  • Identify the pair that sums to 4: (-3 and 7).
This allows us to break down the expression to \( (x + 7)(x - 3) \). Substituting back in, we get the fully factored form of the polynomial from the given exercise.
Polynomials
Polynomials are expressions comprising one or more terms, where each term includes a variable raised to a non-negative integer exponent and has a coefficient.
In the context of the exercise, we worked with a polynomial \(2x^{6} + 8x^{5} - 42x^{4}\), which is a combination of three terms:
  • 2x^{6}
  • 8x^{5}
  • -42x^{4}
Polynomials can be factored by identifying the GCF and breaking the expression into simpler polynomials, as demonstrated. This helps in simplifying expressions and solving polynomial equations more effectively.

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Most popular questions from this chapter

To help you factor the sum or difference of cubes, complete the following list of cubes. $$ \begin{aligned} &1^{3}=\\\ &6^{3}= \end{aligned} $$ $$ \begin{aligned} &2^{3}=\\\ &7^{3}= \end{aligned} $$$$ \begin{aligned} &3^{3}=\\\ &8^{3}= \end{aligned} $$$$ \begin{aligned} &4^{3}=\\\ &9^{3}= \end{aligned} $$$$ \begin{aligned} &5^{3}=\\\ &10^{3}= \end{aligned} $$

Factor completely. \(5 m^{5}+25 m^{4}-40 m^{2}\)

Factor each polynomial. ( Hint: As the first step, factor out the greatest common factor.) $$ 9 x^{2}(r+3)^{3}+12 x y(r+3)^{3}+4 y^{2}(r+3)^{3} $$

Apply the special factoring rules of this section to factor each binomial or trinomial. $$ y^{2}-0.36 $$

Apply the special factoring rules of this section to factor each binomial or trinomial. $$ m^{2}+\frac{2}{3} m+\frac{1}{9} $$
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