Factoring Polynomials: Techniques and Quadratic Formula Application

When faced with the polynomial 4m^2 - 12m - 15, and no Greatest Common Factor (GCF) is apparent, there are methods to proceed. This guide will walk you through the process of factoring such polynomials using the method of factoring by grouping and the quadratic formula if necessary.

Steps to Factor 4m^2 - 12m - 15

1. Identify the Coefficients:

a 4 b -12 c -15

2. Calculate the Product

(text{a} cdot text{c}) 4 (cdot) -15 -60

Find Two Numbers that Multiply to -60 and Add to -12

The numbers 15 and -4 work because (15 cdot -4 -60) and (15 -4 -11).

Rewrite the Middle Term

Replace the middle term, -12m, with 15m - 4m.

4m^2 15m - 4m - 15

Factor by Grouping

Group the terms:

4m^2 15m - 4m - 15

Factor out the common factors in each group:

m(4m 15) - 1(4m 15)

Combine the Groups:

(m - 1)(4m 15)

Conclusion

If you encounter a polynomial that seems hard to factor, check for a GCF first. Then, use the methods described. If these methods fail, you can use the quadratic formula:

(text{x} frac{-text{b} pm sqrt{text{b}^2 - 4text{a}text{c}}}{2text{a}})

This will yield the roots of the polynomial, which can be used to express the polynomial in factored form.

Understanding Polynomial Components

The symbol “^” in a polynomial means to raise the term to the specified exponent or power. The symbols “” and “” mean to multiply, and “/” means to divide. “√” represents the square root of what is in parentheses.

Factors of 4: 1 and 4, -1 and -4, -2 and -2. Factors of -15: 1 and -15, -15 and 1, 3 and -5, 5 and -3.

To determine whether a polynomial has a solution, you can use the quadratic formula:

In ax^2 bx c, the formula is:

(text{x}_1 frac{-text{b} pm sqrt{text{b}^2 - 4text{a}text{c}}}{2text{a}})

This is derived through the process of completing the square. Subtracting “c” on both sides, dividing through by “a”, and then factoring and completing the square leads to the final form.

Given the coefficients of our polynomial, we can apply the quadratic formula to find the roots:

(text{x}_1 frac{-(-12) pm sqrt{(-12)^2 - 4 cdot 4 cdot (-15)}}{2 cdot 4})

(text{x}_1 frac{12 pm sqrt{144 240}}{8})

(text{x}_1 frac{12 pm sqrt{384}}{8})

(text{x}_1 frac{12 pm 4sqrt{24}}{8})

(text{x}_1 frac{12 pm 8.0sqrt{6}}{8})

(text{x}_1 1.5 pm 1.0sqrt{6})

Similarly, (text{x}_2 1.5 - 1.0sqrt{6})

These roots can also be written in factored form:

((x - (1.5 1.0sqrt{6}))(x - (1.5 - 1.0sqrt{6})))

Understanding these techniques and the quadratic formula is crucial for solving more complex polynomial equations.