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Chapter 6: Problem 35

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state. $$9 x^{4}+18 x^{3}+6 x^{2}$$

Short Answer

Expert verified
The factorized form of the polynomial \(9x^4 + 18x^3 + 6x^2\) is \(3x^2(3x^2 + 6x + 2)\).

Step by step solution

01

Identify the Greatest Common Factor

Observe the polynomial \(9x^4 + 18x^3 + 6x^2\) and identify the greatest common factor. Note that the power of \(x\) in every term is at least 2, and you can take out a constant factor of 3 from every term. Therefore, the greatest common factor is \(3x^2\).
02

Factor out the Greatest Common Factor

Now, divide each term of the polynomial by \(3x^2\) since that's the GCF. So, \(9x^4\) divided by \(3x^2\) is \(3x^2\), \(18x^3\) divided by \(3x^2\) is \(6x\), and \(6x^2\) divided by \(3x^2\) is 2. This gives us a new polynomial which is \(3x^2 + 6x + 2\).
03

Write out the Factorized Polynomial

The polynomial \(9x^4 + 18x^3 + 6x^2\), when factorized, becomes \(3x^2(3x^2 + 6x + 2)\).

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Most popular questions from this chapter

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$b x^{2}-4 b+a x^{2}-4 a$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$48 a^{4}-3 a^{2}$$

Exercises 150鈥�152 will help you prepare for the material covered in the next section. Factor: \((x-2)(x+3)-6\)

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$2 x^{4}+6 x^{3} y+2 x^{2} y^{2}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 x z^{2}-72 x z+432 x$$
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