Factor out the greatest common factor from each polynomial. See Example 1. 5(a+3)³-2(a+3)+(a+3)²

Verified step by step guidance

1. Identify the greatest common factor (GCF) in all the terms of the polynomial. Here, each term contains a factor of \((a+3)\) raised to some power.
2. Express each term to clearly show the powers of \((a+3)\): \$5(a+3)^3\(, \)-2(a+3)^1\(, and \)(a+3)^2$.
3. Determine the smallest power of \((a+3)\) present in all terms, which is \((a+3)^1\).
4. Factor out \((a+3)^1\) from each term by dividing each term by \((a+3)\): \(5(a+3)^3 \div (a+3) = 5(a+3)^2\), \(-2(a+3) \div (a+3) = -2\), and \((a+3)^2 \div (a+3) = (a+3)\).
5. Write the factored form as \((a+3)\) multiplied by the resulting polynomial: \((a+3) \left[ 5(a+3)^2 - 2 + (a+3) \right]\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest expression that divides each term of a polynomial without leaving a remainder. Factoring out the GCF simplifies the polynomial and is the first step in many factoring problems. Identifying the GCF involves finding common numerical coefficients and variable expressions shared by all terms.

Factoring Polynomials

Factoring polynomials means rewriting them as a product of simpler expressions. This process often starts by extracting the GCF, which reduces the polynomial into a product of the GCF and a simpler polynomial. Factoring helps in solving equations, simplifying expressions, and analyzing polynomial behavior.

Exponent Rules for Like Bases

When factoring expressions with powers of the same base, such as (a+3)^3, (a+3)^2, and (a+3), understanding exponent rules is essential. The GCF will include the lowest power of the common base, and factoring involves subtracting exponents accordingly. This helps in expressing the polynomial as a product involving powers of the common binomial.