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Chapter 6: Problem 140

Factor: $$ a x^{2}+2 a x+3 a+x^{2}+2 x+3 $$

Short Answer

Expert verified
(a + 1)(x^{2} + 2 x + 3)

Step by step solution

01

- Combine Like Terms

Combine the like terms in the given equation. Group the terms involving the same powers of x:a x^{2} + x^{2} + 2 a x + 2 x + 3 a + 3.
02

- Factor Out Common Factors

Factor out common factors from each group of like terms. First, group the terms:a x^{2} + x^{2} + 2 a x + 2 x + 3 a + 3.Factor out the greatest common factor (GCF) in each group:(x^{2} (a + 1)) + (x (2 a + 2)) + (3 (a + 1)).
03

- Identify Common Binomial Factor

Note that (a + 1) is a common factor:(x^{2} (a + 1) + x (2 a + 2) + 3(a + 1)).
04

- Factor out the Common Binomial

Factor out the common binomial factor (a + 1) from the entire expression:(a + 1)(x^{2} + 2 x + 3).

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Key Concepts

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

Combining Like Terms
Before we can factor the given polynomial, it is crucial to combine like terms. Like terms are terms that have the same variables raised to the same powers. For instance, in the polynomial \(a x^{2} + x^{2} + 2 a x + 2 x + 3 a + 3\), we look for terms involving \(x^{2}\), terms involving \(x\), and constant terms.

Combining these like terms simplifies the polynomial, making it easier to factor. In our case:
  • \(a x^{2}\) and \(x^{2}\) are like terms.
  • \(2 a x\) and \(2 x\) are like terms.
  • \(3 a\) and \(3\) are constant terms.

After combining, the polynomial becomes \(a x^{2} + x^{2} + 2 a x + 2 x + 3 a + 3\). This step streamlines the expression, paving the way for easier factoring.
Greatest Common Factor
Finding and factoring out the Greatest Common Factor (GCF) simplifies polynomial equations. The GCF is the largest factor shared between terms. Let鈥檚 break this down for our polynomial.

First, we group the like terms: \(a x^{2} + x^{2} + 2 a x + 2 x + 3 a + 3\).
Then, we factor out the GCF from each group:
  • For \(a x^{2} + x^{2}\), the GCF is \(x^{2}\).
  • For \(2 a x + 2 x\), the GCF is \(2x\).
  • For \(3 a + 3\), the GCF is \(3\).

This results in: \(x^{2} (a + 1) + x (2 a + 2) + 3 (a + 1)\).
Through this step, the polynomial becomes more structured, allowing us to notice common factors more easily.
Binomial Factors
After factoring out the GCF from each group, it is time to identify and factor out any binomial factors. Binomial factors are expressions that can be factored into the form \((a + b)(c + d)\).

Upon close examination of the polynomial \(x^{2} (a + 1) + x (2 a + 2) + 3 (a + 1)\), we notice that \(a + 1\) is a common binomial.
By factoring out this binomial factor, the expression simplifies to: \((a + 1)(x^{2} + 2 x + 3)\).

This showcases the usefulness of identifying binomial factors鈥攈elping to break down complex polynomials into more manageable factors. Factoring thoroughly can reveal simpler expressions that are easier to work with.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ t^{3}+2 t^{2}-4 t-8 $$

To prepare for Section \(6.5,\) review the product and power rules for exponents and multiplication of polynomials (Sections 5.1 and 5.5 ). Simplify. [ 5.1] $$ \left(-5 x^{2} y\right)^{3} $$

Factor completely. Assume that variables in exponents represent positive integers. $$ a^{2}+2 a b+b^{2}-c^{2}+6 c-9 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ t^{3}+8 t^{2}-t-8 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 8 x^{2}-8 y^{2} $$
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