Problem 22
Question
For the following problems, factor the polynomials. $$ a x^{2}-a $$
Step-by-Step Solution
Verified Answer
Answer: The factored form of the expression is `10(x + 1)`.
1Step 1: Identify the terms of the expression
The given expression is of the form `10x + 10`. This is a binomial, consisting of two terms: `10x`, and `10`.
2Step 2: Find the greatest common factor (GCF) of the terms
For both terms `10x` and `10`, the greatest common factor is `10`.
3Step 3: Factor out the GCF from both terms
Factor out `10` from both terms of the expression by dividing each term by `10` and multiplying the result by `10`. This can be written as:
$$
10(1x + 1)
$$
4Step 4: Write the final factored expression
The factored expression is:
$$
10(x + 1)
$$
Key Concepts
Understanding the Greatest Common FactorNavigating Algebraic ExpressionsBreaking Down Binomials
Understanding the Greatest Common Factor
When looking at algebraic expressions, factoring them can simplify the expression and allow for easier computation or simplification. One key technique in factoring involves identifying the Greatest Common Factor (GCF), which is the largest number that divides evenly into each term of the expression.
To find the GCF, list out the factors of each term and identify the greatest factor that is common to all terms. In our exercise, we observed two terms, 10x and 10. Listing their factors: the factors of 10x are 1, 2, 5, 10, x, and 10x; the factors of 10 are 1, 2, 5, and 10. The greatest factor that both of these share is 10. This crucial step simplifies the process that comes next, which is using the GCF to factor the polynomial.
To find the GCF, list out the factors of each term and identify the greatest factor that is common to all terms. In our exercise, we observed two terms, 10x and 10. Listing their factors: the factors of 10x are 1, 2, 5, 10, x, and 10x; the factors of 10 are 1, 2, 5, and 10. The greatest factor that both of these share is 10. This crucial step simplifies the process that comes next, which is using the GCF to factor the polynomial.
Navigating Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operators such as addition and multiplication. They are used to represent relationships and to solve problems in a variety of mathematical contexts. In factoring algebraic expressions, like our given expression \( 10x + 10 \), you're essentially breaking down a complex expression into simpler, multiplicative components.
Here, after identifying the GCF, the expression is divided by this common factor. In our exercise, both terms are divided by 10, which simplifies the expression to \( x + 1 \). The factored form, \( 10(x + 1) \), is much easier to work with, whether for solving equations, simplifying further, or evaluating for specific values of x.
Here, after identifying the GCF, the expression is divided by this common factor. In our exercise, both terms are divided by 10, which simplifies the expression to \( x + 1 \). The factored form, \( 10(x + 1) \), is much easier to work with, whether for solving equations, simplifying further, or evaluating for specific values of x.
Breaking Down Binomials
A binomial is an algebraic expression that contains exactly two terms, such as \( a + b \) or \( c - d \). Binomials are a fundamental component of algebra and can range from simple to complex. The initial expression in our exercise, \( 10x + 10 \), is a simple binomial.
Factoring binomials involves looking for a GCF that can simplify the expression, as seen in our previous sections. In some cases, binomials can be factored further, especially if they are perfect squares or have special forms, such as the difference of squares. However, in our given example, once the GCF is factored out, we are left with a simplified binomial, emphasizing that factoring is a powerful tool in handling these expressions.
Factoring binomials involves looking for a GCF that can simplify the expression, as seen in our previous sections. In some cases, binomials can be factored further, especially if they are perfect squares or have special forms, such as the difference of squares. However, in our given example, once the GCF is factored out, we are left with a simplified binomial, emphasizing that factoring is a powerful tool in handling these expressions.
Other exercises in this chapter
Problem 21
For the following problems, use the grouping method to factor the polynomials. Some polynomials may not. be factorable using the grouping method. $$ 4 p q-7 p+3
View solution Problem 21
In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ 147
View solution Problem 22
For the following problems, factor the binomials. $$ 2 b^{2}-32 $$
View solution Problem 22
For the following problems, factor the trinomials when possible. $$ x^{2}+6 x-16 $$
View solution