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Chapter 6: Problem 101

Factor. $$ h^{2}(14+r)+5(14+r) $$

Short Answer

Expert verified
The expression factors to \((14+r)(h^2 + 5)\).

Step by step solution

01

Identify Common Factors

In the given expression \( h^{2}(14+r)+5(14+r) \), observe that both terms have \((14+r)\) as a common factor. Hence, we can factor \((14+r)\) out.
02

Factor Out the Common Factor

Factor \((14+r)\) from each term. This gives us:\[(14+r)(h^2 + 5)\]
03

Write the Final Factored Form

After factoring \((14+r)\) out, the expression becomes:\[(14+r)(h^2 + 5)\] This is the fully factored form of the given expression.

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Key Concepts

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

Common Factors
When tackling algebraic expressions, one of the first things you should look for is common factors. A common factor is a number or algebraic term shared by multiple parts of an expression. Identifying common factors is like finding the common thread that connects different parts of the equation.
For example, in the expression \( h^2(14+r)+5(14+r) \), the term \((14 + r)\) is common to both parts of the expression.

Here's how you can spot common factors easily:
  • Look at each term in the expression separately.
  • Ask yourself: "Is there anything shared among the terms?"
  • Focus on both the coefficient and any algebraic terms.
Once you have identified the common factor, you can move on to factor it out, which leads us to our next point.
Factoring Techniques
Factoring is a key algebraic technique used to simplify expressions and solve equations. By taking advantage of common factors, you can break down complicated expressions into simpler parts. Factoring helps in understanding the underlying structure and can be crucial in solving algebraic problems.
In the earlier example \( h^2(14+r)+5(14+r) \), you can apply factoring techniques:
  • First, identify the common factor, which is \((14 + r)\).
  • Next, "factor out" \((14 + r)\) from both terms. This is done by dividing each term by the common factor and placing it outside a parenthesis.
  • What remains inside the parentheses are the terms \(h^2\) and \(5\).
By applying these steps, you make complicated expressions easier to handle and reveal their factored form.
Algebraic Expressions
Algebraic expressions form the backbone of algebra and appear in various math problems. They consist of numbers, variables, and operations like addition and multiplication combined in a meaningful way.
An expression like \( h^2(14+r)+5(14+r) \) is a compact form of representing a relationship. Factoring expressions can often simplify these relationships, making it easier to understand and solve.To deal with algebraic expressions:
  • Understand each term individually, seeing how they combine.
  • Look for common factors which can simplify the entire expression.
  • Don't forget that expressions can often be factored further to reveal simpler parts depending on their structure.
By mastering the art of working with algebraic expressions, you'll improve not just in math classes but in problem-solving skills across various fields.

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Most popular questions from this chapter

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 18 a^{2}-6 a b+42 a c-14 b c $$

Choose the correct method from Sections 6.1, 6.2, or 6.3 to factor each of the following. $$ x^{3}-2 x^{2}+5 x-10 $$

Solve each equation. $$ 2 a^{3}+a^{2}-32 a-16=0 $$

Choose the correct method from Sections 6.1, 6.2, or 6.3 to factor each of the following. $$ 9 x^{4}+27 x^{6} $$

Factor. $$ 8 x^{2}\left(c^{2}+c-2\right)-2 x\left(c^{2}+c-2\right)-\left(c^{2}+c-2\right) $$
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