Factor each polynomial completely. See Example 6. 4x² - 28x + 40

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the coefficients in the polynomial. The coefficients are 4, -28, and 40.
2. Factor out the GCF from the polynomial. In this case, the GCF is 4, so factor 4 out of each term: $4(x^2-7x+10)$.
3. Now, focus on factoring the quadratic expression $x^2-7x+10$. Look for two numbers that multiply to 10 (the constant term) and add to -7 (the coefficient of the linear term).
4. The numbers that satisfy these conditions are -5 and -2. Rewrite the quadratic as $(x-5)(x-2)$.
5. Combine the factored terms with the GCF: $4(x-5)(x-2)$. This is the completely factored form of the polynomial.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions and solving equations. Common methods include factoring out the greatest common factor (GCF), using the difference of squares, and applying the quadratic formula for quadratic polynomials.

Quadratic Polynomials

A quadratic polynomial is a polynomial of degree two, typically expressed in the form ax² + bx + c. The solutions to quadratic equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic polynomials is crucial for effective factoring.

Greatest Common Factor (GCF)

The greatest common factor (GCF) of a set of terms is the largest factor that divides each of the terms without leaving a remainder. Identifying the GCF is often the first step in factoring polynomials, as it allows for simplification by factoring out the GCF, making the remaining polynomial easier to work with.