Problem 31 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ b y^{3}+b y^{2}+b y+b $$

Step-by-Step Solution

Answer

Answer: The factored form of the given polynomial is $(10-3x)(23+2x)$.

Step 1: Identify the common factor

The given polynomial is $$(10-3x)(2+x)+3(10-3x)(7+x).$$ We notice that both terms share a common factor, which is (10-3x).

Step 2: Factor out the common factor

Now we'll factor out the common expression, (10-3x), from both terms: $$(10-3x)[(2+x)+3(7+x)].$$

Step 3: Simplify the expression inside the parentheses

Now, let's simplify the expression inside the square brackets by distributing the 3 to both terms and then combining like terms: $$(10-3x)(2+x+21+3x).$$

Step 4: Simplify and combine like terms

Combine the like terms inside the parentheses: $$(10-3x)(23+2x).$$

Step 5: Present the final factored form

So the factored form of the given polynomial is $$(10-3x)(23+2x).$$

Key Concepts

Common Factors

When factoring polynomials, a crucial initial step is to identify any common factors among the terms involved. In simpler terms, a common factor is a value or expression that appears in each term of a polynomial. Recognizing these early can simplify the problem significantly.

In the problem provided, we can see that both terms of the polynomial, \((10-3x)(2+x)\) and \(3(10-3x)(7+x)\), share a common factor, which is \((10-3x)\). Once identified, this common factor can be factored out, meaning you take it "outside the bracket" of the terms it's attached to.

• This process makes the equation easier to handle and reduces the complexity of subsequent steps.
• Factoring out the common factor first is often very helpful as it simplifies the expressions drastically, making further operations and simplifications possible.

Performing this operation correctly is vital since it lays the foundation for a smooth simplification process.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication). In our problem, expressions such as \((10-3x)\), \((2+x)\), and \((7+x)\) are examples of algebraic expressions formed by different combinations of numbers and variable terms.

Understanding how to manipulate these expressions is essential for successfully factoring and simplifying problems in algebra. This includes knowing how to combine like terms, distribute factors across terms, and the special techniques used to factor them.

• Like terms in algebraic expressions are terms that contain the same variables raised to the same power.
• Always look to simplify expressions by combining like terms wherever possible.

In the given problem, combining like terms after distributing factors plays a critical role in arriving at the final simplified expression.

Distributive Property

The distributive property is a vital algebraic principle that allows us to multiply a single term and two or more terms inside parentheses. It's generally expressed as \(a(b + c) = ab + ac\). Applying this property helps in expanding and simplifying expressions effectively.

In our step-by-step solution, this property was employed when simplifying the expression inside the parentheses: \((10-3x)[(2+x)+3(7+x)]\). Here, we distribute the \(3\) across the terms inside the parentheses:

• Process each term inside by multiplying by the number outside, which was \(3\), resulting in \(3 \times 7 + 3 \times x\).

This step led us further along to combine any like terms, continuing to simplify the polynomial. Understanding when and how to use the distributive property is key to mastering algebra as it simplifies complex expressions and prepares them for more straightforward solutions.