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Chapter 6: Problem 17

An expression is factored when it is written as a product, not a sum. Which of the following are not factored? \(2 k^{2}+(5 k+1)\)

Short Answer

Expert verified
The expression 2k^2 + (5k + 1) is not factored.

Step by step solution

01

Identify terms in the given expression

Start by identifying all the terms in the given expression. The expression provided is 2k^2 + (5k + 1).
02

Check for factors

Determine if the expression is written as a product of factors. A product is formed by multiplying two or more expressions or variables together.
03

Analyze the expression

Analyze if the expression 2k^2 + (5k + 1) can be rewritten as a single product of factors. Since it is in the form of a sum, it cannot be written as a single product of factors without further operations.
04

Conclusion

The expression 2k^2 + (5k + 1) is not factored because it is written as a sum, not as a product of two or more expressions.

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Key Concepts

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

Factoring
Factoring in algebra involves expressing an equation or expression as a product of its factors. This is typically done to simplify expressions and solve equations efficiently.
For instance, consider the expression \( x^2 - 4 \). It can be factored as \( (x - 2)(x + 2) \).
Factoring helps in identifying the roots or solutions of equations. There are different methods of factoring, including:
  • Factoring out the greatest common factor (GCF)
  • Factoring by grouping
  • Factoring trinomials
  • Difference of squares
In the given problem, \( 2k^2 + (5k + 1) \) is a sum and not in a factored form. To factor it, we'd need to rewrite it as a product of two or more expressions.
Algebraic Expressions
Algebraic expressions are mathematical phrases involving numbers, variables (like \( k \) or \( x \)), and operation symbols. They form the core of algebra, enabling us to represent complex mathematical relationships concisely.
Consider the algebraic expression \( 2k^2 + (5k + 1) \). It consists of:
  • Coefficients: These are the numerical parts, like 2 and 5 in the given expression.
  • Variables: These are the letters that stand for unknown numbers, like \( k \) in our expression.
  • Operands: These are the symbols representing mathematical operations, such as + and \(( \) when grouping terms together.
Understanding these components is vital for manipulating and simplifying algebraic expressions.
Polynomials
Polynomials are a type of algebraic expression that consists of terms made up of variables raised to whole number exponents and their coefficients.
For instance, \( 2k^2 + 5k + 1 \) is a polynomial of degree 2 (highest exponent is 2). Each term in a polynomial may include:
  • Monomials: Polynomials with just one term, like \( 2k^2 \).
  • Binomials: Polynomials with two terms, like \( 5k + 1 \).
  • Trinomials: Polynomials with three terms, like \( 2k^2 + 5k + 1 \).
Polynomials are widely used in mathematics to model various real-world scenarios, from physics to economics.
Factoring polynomials simplifies problems, making them easier to solve.

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Most popular questions from this chapter

Brain Busters Factor each polynomial. ( Hint: As the first step, factor out the greatest common factor.) $$ 4 t^{2}(k+9)^{7}+20 t s(k+9)^{7}+25 s^{2}(k+9)^{7} $$

Factor each trinomial completely. See Examples 1鈥�7. ( Hint: In Exercises 55鈥�58, first write the trinomial in descending powers and then factor.) $$ 15 n^{4}-39 n^{3}+18 n^{2} $$

Factor by grouping. \(10 t^{3}-2 t^{2} s^{2}-5 t s+s^{3}\)

Factor each trinomial completely. $$ 9 t^{2}+24 t r+16 r^{2} $$

Factor each trinomial completely. $$ 4 k^{3}-4 k^{2}+9 k $$
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