Problem 137 - Elementary Algebra

Question

Identify the best method to use to factor each polynomial. (a) \(x^{2}+4 x-21\) (b) \(a b+10 b+4 a+40\) (c) \(6 c^{2}+24\)

Step-by-Step Solution

Answer

Factored forms are: (a) \((x + 7)(x - 3)\), (b) \((a + 10)(b + 4)\), (c) \(6(c^{2} + 4)\).

Step 1 - Identify the best method to factor the quadratic polynomial

For the polynomial \(x^{2} + 4x - 21\), use the method of factoring by finding two numbers that multiply to the constant term (-21) and add to the linear coefficient (4).

Step 2 - Factor the quadratic polynomial

The numbers that multiply to -21 and add to 4 are 7 and -3. Hence, the polynomial can be factored as \((x + 7)(x - 3)\).

Step 3 - Identify the best method to factor the four-term polynomial

For polynomial \(ab + 10b + 4a + 40\), use the method of factoring by grouping.

Step 4 - Factor the four-term polynomial

Group the terms in pairs: \((ab + 10b) + (4a + 40)\). Factor out the common factors in each group: \(b(a + 10) + 4(a + 10)\). Now, factor out the common binomial factor \((a + 10)\): \((a + 10)(b + 4)\).

Step 5 - Identify the best method to factor the quadratic polynomial

For \(6c^{2} + 24\), use the method of factoring out the greatest common factor (GCF).

Step 6 - Factor the quadratic polynomial

The GCF of \(6c^{2} + 24\) is 6. Factor out the GCF to get \(6(c^{2} + 4)\).

Key Concepts

quadratic polynomials

Quadratic polynomials are expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. An essential skill in algebra is factoring these polynomials to simplify, solve equations, or find roots.
Common approaches include factoring by:

• Finding two numbers that multiply to the constant term (\(c\)) and add up to the linear coefficient (\(b\)).


• Using the quadratic formula when factoring isn't obvious.

For example, given the quadratic polynomial \(x^2 + 4x - 21\), we need to find two numbers multiplying to \(-21\) and adding to \(4\). These numbers are \(7\) and \(-3\). Therefore, the polynomial can be factored as \((x + 7)(x - 3)\).
Factoring quadratic polynomials effectively expands your toolkit for tackling various algebra problems.

factoring by grouping

Factoring by grouping is a helpful method when dealing with polynomials that have four or more terms. The idea is to group terms with common factors, factor out the commonality, and then combine the factored groups.
Let's look at an example: the polynomial \(ab + 10b + 4a + 40\). Here's the process step-by-step:

• Group the terms: \((ab + 10b) + (4a + 40)\).


• Factor out the common factors in each group: \(b(a + 10) + 4(a + 10)\).


• Notice that \((a + 10)\) is a common factor and factor it out: \((a + 10)(b + 4)\).

Factoring by grouping simplifies complex polynomials, making them easier to work with. This method is particularly useful when direct factoring methods aren't obvious.

greatest common factor

The Greatest Common Factor (GCF) is the largest factor that two or more terms share. Factoring out the GCF from a polynomial is often the first step in simplifying or solving it.
To find the GCF, identify the highest common factor in all terms involved. For instance, consider \(6c^2 + 24\), both terms share a common factor of \(6\). Factoring out the GCF, we get: \(6(c^2 + 4)\). Without factoring out the GCF, solving or simplifying the polynomial would be much more complicated. This step streamlines the process and lays a solid foundation for further factoring or solving equations.
Grasping this concept helps you see patterns and simplify polynomials more intuitively.