Problem 75 Factor completely. If a polynomi... [FREE SOLUTION]

Chapter 6: Problem 75

Factor completely. If a polynomial is prime, state this. $$ a^{4}+8 a^{2}+8 a^{3}+64 a $$

Short Answer

Expert verified

The factored form is (a^2 + 8)(a^2 + 8a). Neither can be factored further.

Step by step solution

Group Terms

Start by grouping the polynomial into parts that can be factored easily. Group the terms into pairs: Group a^{4} + 8a^{2} + 8a^{3} + 64a can be rearranged to (a^4 + 8a^2) + (8a^3 + 64a)

Factor Out the Greatest Common Factor (GCF)

Factor out the GCF from each pair: The GCF of the first pair (a^4 + 8a^2) is: a^2 so, a^{4} + 8a^{2} = a^{2}(a^{2} + 8) The GCF of the second pair (8a藛3 + 64a) is: 8a so, 8a^{3} + 64a = 8a(a^{2} + 8)

Combine the Factored Groups

Combine the factored groups: The polynomial now can be written as: a^{2}(a^{2} + 8) + 8a(a^{2} + 8)

Factor Out the Common Binomial Factor

Factor out the common binomial factor: (a^{2} + 8) which gives: (a^{2} + 8)(a^{2} + 8a)

Check for Further Factorization

What remains to be factored is: (a^2 + 8)(a^2 + 8a) Notice that neither (a^2 + 8) nor (a^2 + 8a) can be factored further as simple polynomials or other expressions

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Key Concepts

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

grouping terms

Grouping is an essential first step in factoring polynomials. It involves rearranging and pairing terms in a way that makes the polynomial easier to factor.
For example, in our given polynomial $a^4 + 8a^2 + 8a^3 + 64a$, we can rearrange it as $(a^4 + 8a^2) + (8a^3 + 64a)$.
This step helps to simplify the polynomial by breaking it down into manageable parts.

greatest common factor

The Greatest Common Factor (GCF) is the largest value that divides all terms in a group or a polynomial.
In the first group $(a^4 + 8a^2)$, the GCF is $a^2$. So, we factor it out: $a^2(a^2 + 8)$.
In the second group $(8a^3 + 64a)$, the GCF is $8a$. Thus, we factor it out: $8a(a^2 + 8)$.
By factoring the GCF from each group, we simplify the polynomial and prepare it for further factorization.

binomial factorization

Binomial factorization involves factoring expressions that contain two terms into their simplest form.
In our example, after we factor out the GCFs, we get $a^{2}(a^2 + 8) + 8a(a^2 + 8)$.
Notice that $(a^2 + 8)$ is a common binomial factor in both groups.
This means we can factor $(a^2 + 8)$ out from the entire expression: $(a^2 + 8)(a^2 + 8a)$.
Recognizing common binomials is key to simplifying complex polynomials.

polynomial factorization steps

Let's summarize the general steps for factoring polynomials:

Step 1: Group the terms to simplify the polynomial.
Step 2: Find and factor out the GCF from each group.
Step 3: Combine the factored groups.
Step 4: Identify and factor out any common binomial factors.
Step 5: Check if the resulting factors can be factored further.

By following these steps, we can systematically break down and factor even complicated polynomials.

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91影视

Factor completely. If a polynomial is prime, state this. $$ a^{4}+8 a^{2}+8 a^{3}+64 a $$

Short Answer

Step by step solution

Group Terms

Factor Out the Greatest Common Factor (GCF)

Combine the Factored Groups

Factor Out the Common Binomial Factor

Check for Further Factorization

Key Concepts

grouping terms

greatest common factor

binomial factorization

polynomial factorization steps

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