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Chapter 6: Problem 38

$$ 30 x^{3}-155 x^{2}+25 x $$

Short Answer

Expert verified
Factor the expression: \(x(30x^2 - 155x + 25)\).

Step by step solution

01

Identify Common Factor

Look at all the terms in the expression: \(30x^3, -155x^2, \) and \(25x\). Notice that each term has \(x\) in it, so we can factor out an \(x\) from the entire expression.
02

Factor Out the Common Factor

Factor \(x\) from each term of the expression. This gives: \[x(30x^2 - 155x + 25)\]
03

Simplify Remaining Expression

Look at the quadratic \(30x^2 - 155x + 25\). Now factor this trinomial. Try factor pairs of 30 (e.g., 1 and 30, 2 and 15, etc.) that combine by addition or subtraction to get -155.

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Key Concepts

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

Common Factor
Identifying a common factor is the first step in making algebraic equations simpler. A common factor is a number or variable that is present in each term of an algebraic expression. Math problems like \(30x^3 - 155x^2 + 25x\) may seem complex, but by finding a common factor, you can make them simpler to manage.
  • In our example, the terms are \(30x^3\), \(-155x^2\), and \(25x\). Here, the variable \(x\) is a common factor because it is present in all terms.
  • By factoring out \(x\), you rewrite the expression as \(x(30x^2 - 155x + 25)\), which is more manageable.
Knowing how to identify and extract common factors simplifies expressions and makes further operations more straightforward.
Quadratic Expression
A quadratic expression takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our case, after factoring out the common variable \(x\), we're left with a quadratic expression: \(30x^2 - 155x + 25\).
  • The "\(a\)" in this example is 30, "\(b\)" is -155, and "\(c\)" is 25.
  • Quadratic expressions are solved and simplified using methods such as factorization, completing the square, or using the quadratic formula.
Understanding the structure and characteristics of a quadratic expression is crucial in further solving, simplifying, or graphing these equations.
Trinomial Factorization
Trinomial factorization involves breaking down a quadratic expression into simpler expressions that multiply to make the original trinomial. Given \(30x^2 - 155x + 25\), the goal is to find two binomials that multiply together to form the original expression.
  • Start by identifying factor pairs of the leading coefficient (30) and the constant term (25) to find combinations that can add up or subtract to the middle term (-155).
  • This might involve a bit of trial and error and checking several combinations before finding the right pair.
  • Once the correct pair of numbers is found, rewrite the middle term \(-155x\) using the pair, then group the terms, factor by grouping, and write the product of two binomials.
Trinomial factorization is a vital skill in algebra that helps simplify and solve complex quadratic equations by breaking them into easier-to-manage parts.

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Most popular questions from this chapter

Factor each perfect square trinomial completely. See Examples 8 through 11. \(16 y^{2}+40 y+25\)

Factor each four-term polynomial by grouping. See Examples 11 through 16. $$ 5 m^{3}+6 m n+5 m^{2}+6 n $$

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1 ). Don't forget to factor out the GCF first. See Examples I through 10. $$ 5 x^{3} y-25 x^{2} y^{2}-120 x y^{3} $$

Factor a negative number or a GCF with a negative coefficient from each polynomial. See Example 5. $$ -5 m^{6}+10 m^{5}-5 m^{3} $$

Factor completely. See Examples 1 through 7 . \(12 x^{3}+11 x^{2}+2 x\)
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