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Chapter 6: Problem 26

Factor completely. $$ -18 y^{2}+y^{3}+81 y $$

Short Answer

Expert verified
The factored form is \( y(y - 9)^2 \).

Step by step solution

01

Rewrite the expression in standard form

First, rewrite the given expression in standard polynomial form by arranging the terms in descending order of their degree. The expression becomes: \( y^3 - 18y^2 + 81y \).
02

Factor out the greatest common factor (GCF)

Identify and factor out the greatest common factor (GCF) from all terms. Here, each term has a factor of \( y \). So, factor out the GCF: \( y(y^2 - 18y + 81) \).
03

Factor the quadratic expression

Next, factor the quadratic expression \( y^2 - 18y + 81 \). This expression is a perfect square trinomial, because \( y^2 - 18y + 81 = (y - 9)^2 \).
04

Combine the factors

Write the factored form by combining the factored out GCF with the factored quadratic expression: \( y(y - 9)^2 \).

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Key Concepts

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

polynomial expressions
Polynomial expressions are mathematical phrases that involve variables, coefficients, and exponents. These expressions can have multiple terms. In our original exercise, the polynomial expression is given as \( -18y^2+y^3+81y \).
An important aspect when dealing with polynomial expressions is to arrange them in standard form.
Standard form means writing terms in descending order of their degree, which is the highest power of the variable in each term.
When rearranged, \( -18y^2+y^3+81y \) becomes \( y^3-18y^2+81y \). This helps in identifying patterns and factors more easily.
greatest common factor
The greatest common factor (GCF) is the largest factor that divides all the terms in a polynomial. To find the GCF, identify the highest common variable and numerical factor among each term.
Here鈥檚 how we apply it to our example:
  • Look at the polynomial \( y^3 - 18y^2 + 81y \).
  • Each term includes a \( y \).
So, \( y \) is the GCF. By factoring out \( y \), we simplify our expression:
\( y(y^2 - 18y + 81) \). By identifying and factoring out the GCF first, solving the remaining expression becomes easier.
perfect square trinomials
A perfect square trinomial is a special type of quadratic that forms when a binomial is squared. These trinomials are of the form: \( a^2 + 2ab + b^2 \) or \( a^2 - 2ab + b^2 \).
In our expression, we have \( y^2 - 18y + 81 \). Let鈥檚 break it down:
  • \( y^2 \) is a square term.
  • \( 81 \) is another square term, specifically \( 9^2 \).
  • \( -18y \) is twice the product of \( y \) and \( 9 \) (i.e., \( 2(y)(-9) \))
Thus, \( y^2 - 18y + 81 \) is a perfect square trinomial and can be written as: \( (y - 9)^2 \). Substituting back, the original expression fully factored is: \( y(y - 9)^2 \).

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Most popular questions from this chapter

Review multiplying binomials using FOIL $$ (t-5)(t+10) $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 2 a^{4}-32 y^{8} $$

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