Problem 56
Question
Factor by grouping. $$a x^{2}-2 x^{2}+3 a-6$$
Step-by-Step Solution
Verified Answer
The factored form is \((x^2 + 3)(a - 2)\).
1Step 1: Identify Pairs for Grouping
Group the terms in the given expression to facilitate factoring. The expression is \( ax^2 - 2x^2 + 3a - 6 \). We can group them as \((ax^2 - 2x^2) + (3a - 6)\).
2Step 2: Factor the Common Factors from Each Pair
For the first pair, \( ax^2 - 2x^2 \), factor out \( x^2 \): \( x^2(a - 2) \). For the second pair, \( 3a - 6 \), factor out \( 3 \): \( 3(a - 2) \).
3Step 3: Combine Factored Expressions
The expression can now be combined since both groupings contain the common binomial factor \( (a - 2) \): \( (x^2 + 3)(a - 2) \).
4Step 4: Verify the Factored Expression
Expand the factored expression to ensure it matches the original expression. Expanding \( (x^2 + 3)(a - 2) \) gives \( ax^2 - 2x^2 + 3a - 6 \), which matches the original expression.
Key Concepts
Polynomial ExpressionsCommon FactorsBinomial FactorAlgebraic Manipulation
Polynomial Expressions
Polynomial expressions are mathematical phrases that involve a sum of powers of one or more variables, multiplied by coefficients. For example, in the expression \( ax^2 - 2x^2 + 3a - 6 \), each term is a part of this polynomial.
Polynomials can have various numbers of terms, sometimes classified as monomials, binomials, or trinomials, depending on if there are one, two, or three terms respectively.
In this context, the expression consists of four terms, making it a polynomial with four distinct elements.
Polynomials can have various numbers of terms, sometimes classified as monomials, binomials, or trinomials, depending on if there are one, two, or three terms respectively.
In this context, the expression consists of four terms, making it a polynomial with four distinct elements.
Common Factors
Identifying common factors is a crucial step when factoring polynomial expressions. A common factor is a number, variable, or phrase, shared by terms in an expression.
In the example \( ax^2 - 2x^2 + 3a - 6 \), we can group the terms to identify common factors more easily.
In the example \( ax^2 - 2x^2 + 3a - 6 \), we can group the terms to identify common factors more easily.
- For the group \( ax^2 - 2x^2 \), both terms share the factor \( x^2 \).
- For the second group \( 3a - 6 \), \( 3 \) is a common factor.
Binomial Factor
A binomial factor is composed of two terms, each of which may include numbers, variables, or both. When factoring by grouping, our goal often involves finding a common binomial factor after grouping the terms.
In our expression \( ax^2 - 2x^2 + 3a - 6 \), by the time we factor out the common factors, we end up with expressions \( x^2(a - 2) \) and \( 3(a - 2) \).
Here, \( (a - 2) \) emerges as the common binomial factor, present in both part of the expression. This allows us to extract this binomial factor and create a product involving it, which greatly simplifies the polynomial.
In our expression \( ax^2 - 2x^2 + 3a - 6 \), by the time we factor out the common factors, we end up with expressions \( x^2(a - 2) \) and \( 3(a - 2) \).
Here, \( (a - 2) \) emerges as the common binomial factor, present in both part of the expression. This allows us to extract this binomial factor and create a product involving it, which greatly simplifies the polynomial.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions to make them easier to interpret or solve. It requires a deep understanding of algebraic properties and operations.
When working through the expression \( ax^2 - 2x^2 + 3a - 6 \), we utilized grouping and factored out common terms through algebraic manipulation.
Steps include:
When working through the expression \( ax^2 - 2x^2 + 3a - 6 \), we utilized grouping and factored out common terms through algebraic manipulation.
Steps include:
- Grouping terms to identify similar patterns or factors.
- Factoring out commonalities, such as numbers or variables, to simplify the expression.
- Combining the factored sections through multiplication of the common binomial factor, leading to a simplified expression \( (x^2 + 3)(a - 2) \).
Other exercises in this chapter
Problem 56
Factor completely each of the polynomials and indicate any that are not factorable using integers. $$x^{4}-13 x^{2}+36$$
View solution Problem 56
Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$x^{6}+y^{6}$$
View solution Problem 56
Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(x+6)\left(2 x^{2}-x
View solution Problem 56
Raise each monomial to the indicated power. $$\left(-4 a b c^{4}\right)^{3}$$
View solution