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Chapter 6: Problem 255

Factor completely. $$ 4 x^{2}+40 x+84 $$

Short Answer

Expert verified
4(x + 3)(x + 7)

Step by step solution

01

- Identify the Greatest Common Factor (GCF)

First, find the greatest common factor of the coefficients 4, 40, and 84. The GCF is 4.
02

- Factor out the GCF

Rewrite the polynomial by factoring out the GCF (which is 4) from each term: \[ 4(x^2 + 10x + 21) \]
03

- Factor the quadratic expression

Next, factor the quadratic expression inside the parentheses. Find two numbers that multiply to 21 (the constant term) and add up to 10 (the coefficient of the linear term). These numbers are 3 and 7: \[ x^2 + 10x + 21 = (x + 3)(x + 7) \]
04

- Combine the factors

Combine the factored terms with the GCF: \[ 4(x + 3)(x + 7) \]

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Key Concepts

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

greatest common factor
When factoring polynomials, finding the Greatest Common Factor (GCF) is essential. The GCF is the largest number that divides each term of the polynomial without any remainder.
In the exercise, the given polynomial is: 4x虏 + 40x + 84.
To identify the GCF of the coefficients (4, 40, and 84), we start by finding the prime factors of each:
  • 4 = 2虏
  • 40 = 2鲁 脳 5
  • 84 = 2虏 脳 3 脳 7
The common prime factor between these coefficients is 2虏. Thus, the GCF of 4, 40, and 84 is 4.
Next, we factor out the GCF from the polynomial: 4(x虏 + 10x + 21). By doing this step, the polynomial becomes easier to work with and prepares it for further factoring.
quadratic expression
A quadratic expression is a polynomial of degree 2, in the form: ax虏 + bx + c. In the exercise, after factoring out the GCF, we get the quadratic expression: x虏 + 10x + 21.
The goal is to factor this expression further. We need to find two numbers that both multiply to the constant term (21) and add up to the coefficient of the linear term (10). These numbers are 3 and 7. We can write: x虏 + 10x + 21 = (x + 3)(x + 7).
This step makes use of the property that (a + b)(a + c) expands to a虏 + (b + c)a + bc. Here a = x, b = 3, and c = 7.
factoring techniques
Factoring techniques are methods used to simplify polynomials. One essential technique is factoring out the Greatest Common Factor. Other techniques include:
  • Factoring by grouping
  • Factoring trinomials
  • Difference of squares
In the given exercise, after determining the GCF and factoring it out, we used the technique of factoring a quadratic trinomial. This process involves finding two numbers that multiply to the constant term and add to the middle term. To summarize:
It starts with: 4(x虏 + 10x + 21).
Then, it factors further to: 4(x + 3)(x + 7).
This final expression represents the completely factored form of the original polynomial: 4x虏 + 40x + 84.

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Most popular questions from this chapter

In the following exercises, factor each expression using any method. $$ 3 q^{2}+6 q+2 $$

In the following exercises, factor each expression using any method. $$ 3 p^{2}+21 p $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x y+c y^{2} .\) $$ x^{2}-2 x y-80 y^{2} $$

In the following exercises, factor completely using trial and error. $$ 5 y^{2}+16 y+11 $$

In the following exercises, foctor using substitution. $$ x^{4}-3 x^{2}-28 $$
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