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Chapter 6: Problem 28

Factor out the greatest common monomial factor from the polynomial. $$ 9-3 y+15 y^{2} $$

Short Answer

Expert verified
The greatest common monomial factor factored from the polynomial \(9 - 3y + 15y^{2}\) is 3, and the resulting polynomial is \(3(3 - y + 5y^{2})\).

Step by step solution

01

Identify the coefficients and variables

First, list out the coefficient or number part of each term in the polynomial. We can see the coefficients are 9, -3, and 15. Also, note that the variable 'y' appears in two of the three terms.
02

Find the greatest common factor of the coefficients

Next, identify the greatest number that divides evenly into 9, -3, and 15, which is 3. This is the greatest common factor of the coefficients.
03

Factor out the greatest common factor

Now, divide each term of the polynomial \(9 - 3y + 15y^{2}\) by 3. The new expression will be \(3(3 - y + 5y^{2})\). This expression is the factorized form of the original polynomial, where 3 is the greatest common factor.

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Key Concepts

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

Factoring Polynomials
Polynomials can often look tricky, but factoring them makes them a lot easier to handle. Factoring involves breaking down a polynomial into simpler parts called factors. These factors, when multiplied together, will give you the original polynomial. In our context, we're aiming to find the greatest common factor among the terms, which we will use to simplify the polynomial.
To factor a polynomial:
  • Identify common factors in all terms.
  • Factor them out, leaving behind a simpler expression.
For example, consider the polynomial \( 9 - 3y + 15y^2 \). First, look for a common factor in all terms and then divide each term by this common factor. This reveals a more straightforward polynomial that often provides better insights into the structure or solution of the problem.
Greatest Common Divisor
The greatest common divisor (GCD) is a smart way to tackle the complexity in your polynomial. It's the largest number that can exactly divide each of the terms in the polynomial without leaving a remainder. Finding the GCD of the coefficients will help you identify what can be factored out easily.
In our example:
  • Coefficients are 9, -3, and 15.
  • The GCD here is 3, as it divides all three numbers without a remainder.
Understanding the GCD is crucial because it simplifies calculations. Once you factor out the GCD from the polynomial, you are left with a simpler expression, making it much easier to work with. It's like trimming down elements to their simplest form!
Coefficient Identification
In dealing with polynomials, it's essential to understand each term's coefficient, which is the numerical part multiplying the variable. Coefficients provide necessary information on how terms relate to each other through their size and value.
Let's breakdown this method:
  • Examine each term individually.
  • Identify the numerical coefficients, such as 9 in the first term of our polynomial example \( 9 - 3y + 15y^2 \).
  • Observing the coefficients helps highlight similarities or common factors across different terms.
By carefully determining these coefficients, you set the groundwork for efficiently finding the greatest common divisor, which subsequently aids in factoring. Proficient coefficient identification simplifies polynomial expressions and is a pivotal step in solving polynomial equations effectively.

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Most popular questions from this chapter

In Exercises 67-74, factor the polynomial completely. $$ \frac{9}{16}-x^{2} $$

Solve the equation. $$ 2 y^{2}-y=3 $$

Determine whether the statement is true or false. Justify your answer. The only equation with solutions \(x=2\) and \(x=-5\) is \((x-2)(x+5)=0 .\)

Use the Zero-Factor Property to solve the equation. $$ z(z-3)=0 $$

Is the equation \(2 x^{2}+18 x=0\) in general form? Explain.
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