In Exercises 1–10, factor out the greatest common factor. 9x^4−18x^3+27x^2

Verified step by step guidance

1. Step 1: Identify the greatest common factor (GCF) of the terms in the expression. In this case, the GCF is 9x^2, because it is the highest power of x that divides evenly into all terms and 9 is the largest number that divides evenly into the coefficients of all terms.
2. Step 2: Rewrite the expression as the product of the GCF and the remaining factors. This can be done by dividing each term in the original expression by the GCF. So, the expression becomes 9x^2(x^2 - 2x + 3).
3. Step 3: Check your work by distributing the GCF back into the parentheses. If you get the original expression, then you have factored out the GCF correctly.
4. Step 4: The factored form of the expression is 9x^2(x^2 - 2x + 3).
5. Step 5: Remember that factoring is the process of breaking down an expression into its simplest parts. It's a useful skill in algebra that can help simplify complex expressions and solve equations.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest integer or algebraic expression that divides each term of a polynomial without leaving a remainder. To find the GCF, identify the common factors in the coefficients and the variables of each term. For example, in the expression 9x^4, -18x^3, and 27x^2, the GCF is 9x^2, as it is the highest factor that can be factored out from all terms.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process simplifies expressions and can make solving equations easier. In the case of the polynomial 9x^4 - 18x^3 + 27x^2, factoring out the GCF allows us to express it in a simpler form, which can then be further analyzed or solved.

Polynomial Terms

Polynomial terms are the individual components of a polynomial, typically expressed in the form ax^n, where 'a' is a coefficient, 'x' is a variable, and 'n' is a non-negative integer exponent. Understanding the structure of polynomial terms is crucial for factoring, as it helps identify common factors and simplifies the polynomial. In the given expression, the terms are 9x^4, -18x^3, and 27x^2.