Problem 13 The following exercises are of m... [FREE SOLUTION]

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Chapter 5: Problem 13

The following exercises are of mixed variety. Factor each polynomial. $$ 4(p+2)+m(p+2) $$

Short Answer

Expert verified

\[ (p+2)(4+m) \]

Step by step solution

Identify Common Factor

Both terms in the polynomial share a common factor $p+2$. Identify this common factor.

Factorize Using Common Factor

Factor out the common factor $p+2$ from the expression. This gives us: \[ (p+2)(4+m) \]

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Key Concepts

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

Common Factor

To start with any factorization problem, identifying the common factor is crucial. A common factor is a term or expression that appears in every term of a polynomial.
This means it鈥檚 something that you can evenly divide each part of the polynomial by.

Let's look at the given polynomial:

\[ 4(p+2) + m(p+2) \] Notice that both terms in the polynomial, 4(p+2) and m(p+2), have a common piece they share, which is (p+2).

Identifying common factors helps us simplify expressions and is always the first key step in factorization.

Factorization

Factorization involves rewriting a polynomial as a product of simpler factors. This makes it easier to work with in future algebraic operations.

Once you've found your common factor, you can extract it from the polynomial. With our expression:

\[ 4(p+2) + m(p+2) \]
Since (p+2) is common to both terms, we factor it out to simplify the expression.
This results in:

\[ (p+2)(4+m) \] This means we've rewritten our original polynomial as a product of two factors.

Factoring out the common factor breaks down the complex polynomial into simpler components, making further algebraic manipulations much more manageable.

Polynomial Expressions

Polynomials are expressions involving variables and coefficients, combined using addition, subtraction, and multiplication. They do not include division by a variable.

For instance, let's consider our original expression:

\[ 4(p+2) + m(p+2) \]
Polynomials are a foundation in algebra and come in different forms, like monomials, binomials, and trinomials.
In our task, we had a binomial, which is a polynomial with exactly two terms.

Working with polynomial expressions becomes easier when you understand how to break them down into factors. The simpler components can then be analyzed and solved more easily.

Algebraic Techniques

Algebraic techniques are methods used to manipulate mathematical expressions and solve equations.
Factorization is just one of these essential techniques.

Other common algebraic techniques include:

Simplifying expressions
Expanding polynomials
Using the distributive property
Solving equations

In our exercise, we used the distributive property in reverse to factor out the common factor.
This is essential because many algebraic problems require rearranging and simplifying expressions before solving them.

Practicing these techniques helps build your problem-solving skills and makes tackling more complex algebraic challenges less daunting.
Remember, the more you practice, the more confident you'll become in handling any algebraic expressions.

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91影视

The following exercises are of mixed variety. Factor each polynomial. $$ 4(p+2)+m(p+2) $$

Short Answer

Step by step solution

Identify Common Factor

Factorize Using Common Factor

Key Concepts

Common Factor

Factorization

Polynomial Expressions

Algebraic Techniques

One App. One Place for Learning.

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