Question

Factor each polynomial. $$ 5 x^{2}+22 x+8 $$

Short answer

\( (x + 4)(5x + 2) \)

Step-by-step solution

- Identify Coefficients

Identify the coefficients in the polynomial. The polynomial is written as \(a x^{2}+b x+c\). Here, \(a = 5\), \(b = 22\), and \(c = 8\).

- Find Two Numbers

Find two numbers that multiply to \(a*c = 5*8 = 40\) and add up to \(b = 22\). The numbers that satisfy both conditions are 2 and 20.

- Rewrite the Middle Term

Rewrite the middle term using the two numbers found: \(22x = 2x + 20x\). The polynomial now becomes: \[5x^2 + 2x + 20x + 8\].

- Factor by Grouping

Group terms to make factoring easier: \[(5x^2 + 2x) + (20x + 8)\]. Factor out the greatest common factor (GCF) from each group: \[x(5x + 2) + 4(5x + 2)\].

- Factor Out Common Binomial

Factor out the common binomial factor \((5x + 2)\): \[(x + 4)(5x + 2)\].

Key concepts

Polynomial Coefficients

Understanding polynomial coefficients is crucial in the process of factoring polynomials. A polynomial is usually written in the form of:

Here, the letters 'a', 'b', and 'c' are known as coefficients:

• 'a' is the coefficient of the quadratic term () which is 5 in our case.
• 'b' is the coefficient of the linear term () which is 22.
• 'c' is the constant term, which is 8.

Identifying these coefficients correctly is the first step towards factoring a polynomial.

Factoring by Grouping

Factoring by grouping is a method used when a polynomial has four terms, as we will create here to ease our factoring process. First, we find two numbers that multiply to the product of 5 (the coefficient of ) and 8 (the constant term). These two numbers are 2 and 20 because:

• 2 x 20 = 40 (which matches ).
• 2 + 20 = 22 (which matches 'b').

Then, rewrite the polynomial:
<5x^2 + 2x + 20x + 8>
Now, grouping terms to find more manageable factor pairs. This looks like:
<(5x^2 + 2x) + (20x + 8)>
Each parenthesis can be factored by their greatest common factor (GCF).

Greatest Common Factor

The Greatest Common Factor (GCF) plays a significant role in simplifying expressions. For each group within the parentheses, we identify and factor out the GCF:

• In the first group <5x^2 + 2x>, the GCF is 'x', so we factor it as:
  
• In the second group <20x + 8>, the GCF is 4, so we factor it as:
  <4(5x + 2)>

Notice, that both groups have a common binomial factor, <(5x + 2)>.
So the polynomial can be factored as:
<(x + 4)(5x + 2)>. This is our factored expression, and the exercise is complete!