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Chapter 6: Problem 57

Factor completely. $$ 2 x^{6}+8 x^{5}-42 x^{4} $$

Short Answer

Expert verified
The complete factorization is \[ 2x^4 (x + 7)(x - 3) \].

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of all the terms. In this case, all the coefficients (2, 8, and -42) share a common factor of 2, and all terms include at least an \( x^4 \). So, the GCF is \( 2x^4 \).
02

Factor out the GCF

Divide each term by the GCF \( 2x^4 \) and rewrite the expression as the product of the GCF and the simplified polynomial: \[ 2x^6 + 8x^5 - 42x^4 = 2x^4(x^2 + 4x - 21) \].
03

Factor the Quadratic Expression

Now, factor the quadratic expression \( x^2 + 4x - 21 \) inside the parentheses. Look for two numbers that multiply to -21 and add to 4. These numbers are 7 and -3. Thus, we can write: \[ x^2 + 4x - 21 = (x + 7)(x - 3) \].
04

Combine All the Factors

Combine the GCF with the factored quadratic expression to get the complete factorization: \[ 2x^6 + 8x^5 - 42x^4 = 2x^4 (x + 7)(x - 3) \].

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Key Concepts

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

Greatest Common Factor
Factoring starts with finding the Greatest Common Factor (GCF). The GCF is the largest number and the highest power of any variables that can divide each term of a polynomial without any remainder. For example, in the polynomial \(2x^6 + 8x^5 - 42x^4\), identify the GCF by looking at both the numerical coefficients and the variables. The coefficients \(2, 8,\) and \(-42\) share a common factor of \(2\). All terms also include at least \(x^4\). Hence, the GCF is \(2x^4\). This helps to simplify the polynomial and makes further factoring easier.
Factoring Quadratic Expressions
Once the GCF is factored out, the next step involves factoring any quadratic expression within the polynomial. A quadratic expression is in the form \(ax^2 + bx + c\). For instance, after factoring out \(2x^4\) from \(2x^6 + 8x^5 - 42x^4\), we are left with \(x^2 + 4x - 21\). To factor this, find two numbers that multiply to the constant term \(-21\) (the product) and add to the coefficient of the middle term \(4\) (the sum). These numbers are \(7\) and \(-3\), hence:
\(x^2 + 4x - 21 = (x + 7)(x - 3)\). This factorization splits the quadratic into two binomial expressions, making them easier to handle.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. The entire process combines finding the GCF and factoring other expressions within the polynomial.
For the example polynomial \(2x^6 + 8x^5 - 42x^4\):
  • Step 1: Identify and factor out the GCF, which is \(2x^4\).
  • Step 2: After factoring out \(2x^4\), you get \(2x^4 (x^2 + 4x - 21)\).
  • Step 3: Factor the quadratic \(x^2 + 4x - 21\) to get \((x + 7)(x - 3)\).
  • Step 4: Combine all factors to complete the factorization: \(2x^6 + 8x^5 - 42x^4 = 2x^4 (x + 7)(x - 3)\).
This methodical approach ensures accurate factorization, simplifying the polynomial for further applications.

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Most popular questions from this chapter

Factor each polynomial. $$ a^{5}+3 a^{4} b-4 a^{3} b^{2} $$

Factor by grouping. \(4 x^{3}+3 x^{2} y+4 x y^{2}+3 y^{3}\)

Factor completely. If the polynomial cannot be factored, write prime. $$ -32+14 x+x^{2} $$

Factor each trinomial completely. $$ 3 y^{2}+48 y+192 $$

When a student was directed to factor \(k^{4}-81\) from Exercise 30 completely, his teacher did not give him full credit for the answer $$ \left(k^{2}+9\right)\left(k^{2}-9\right) $$
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