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Chapter 6: Problem 23

Factor completely. If a polynomial is prime, state this. $$ 10-23 x+12 x^{2} $$

Short Answer

Expert verified
-2a^4(a - 2)^2

Step by step solution

01

- Identify the Greatest Common Factor (GCF)

First, determine if there is any common factor in all the terms of the polynomial. The terms are -2a^6, 8a^5, and -8a^4. The numerical GCF is 2, and the variable GCF is a^4. Therefore, the overall GCF is -2a^4.
02

- Factor out the GCF

Next, factor out the GCF -2a^4 from each term of the polynomial. This gives: -2a^4(a^2 - 4a + 4)
03

- Simplify the Remaining Polynomial

Now, look at the remaining polynomial inside the parentheses: a^2 - 4a + 4. This is a quadratic expression. To factor this, note that it is a perfect square trinomial: (a - 2)^2.
04

- Combine the Factors

Combine the factors to get the fully factored form of the polynomial: -2a^4(a - 2)^2

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) is the largest factor that divides two or more terms without a remainder. Identifying the GCF is a crucial step when factoring polynomials as it simplifies the polynomial significantly.
In our given polynomial -2a^6, 8a^5, and -8a^4, we need to first find the GCF of the numerical coefficients and the variable parts separately:
  • For the numerical parts: -2, 8, and -8, the GCF is 2.
  • For the variable parts: a^6, a^5, and a^4, the GCF is a^4.
Combining these, the overall GCF of the polynomial is -2a^4.
Factoring this out from every term results in -2a^4(a^2 - 4a + 4), simplifying our expression and making further factoring easier.
Quadratic Expression
A quadratic expression is a polynomial of degree 2, typically written in the standard form as ax^2 + bx + c. Understanding how to identify and factor quadratic expressions is essential when simplifying polynomials.
In the context of our problem, after factoring out the GCF, we are left with a quadratic expression inside the parentheses: a^2 - 4a + 4.
The standard form for this expression is a^2 - 4a + 4, where:
  • a = 1
  • b = -4
  • c = 4.
To factor this quadratic expression, we look for two numbers that multiply to c (4) and add up to b (-4).
In this case, -2 and -2 are those numbers, and thus the quadratic can be factored into (a - 2)(a - 2), or (a - 2)^2, showcasing it's a perfect square trinomial.
Perfect Square Trinomial
A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. This form is beneficial because it streamlines the factoring process.
Recognizing a perfect square trinomial is key; it generally takes the form a^2 卤 2ab + b^2.
In our given polynomial a^2 - 4a + 4, we notice:
  • a^2 is the square of 'a'
  • 4 is the square of '2'
  • -4a is twice the product of 'a' and '2' (-2a).
Thus, this trinomial fits the pattern of a perfect square.
Therefore, a^2 - 4a + 4 can be rewritten as (a - 2)^2, as explained earlier.
By identifying it as a perfect square trinomial, we simplify the factoring process significantly, leading to the complete factorized form -2a^4(a - 2)^2.

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Most popular questions from this chapter

The square of a number plus the number is \(132 .\) Find all such numbers.

A rectangular garden is \(30 \mathrm{ft}\) by \(40 \mathrm{ft}\). Part of the garden is removed in order to install a walkway of uniform width around it. The area of the new garden is one-half the area of the old garden. How wide is the walkway?

Solve. Round any irrational solutions to the nearest thousandth. $$ 2 x^{2}+8 x+1=0 $$

Write an equivalent expression by factoring out the greatest common factor. $$ 9 x^{3} y^{6} z^{2}-12 x^{4} y^{4} z^{4}+15 x^{2} y^{5} z^{3} $$

To prepare for Section \(6.5,\) review the product and power rules for exponents and multiplication of polynomials (Sections 5.1 and 5.5 ). Multiply.[ 5.5]. $$ (x+1)(x+1)(x+1) $$
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