Problem 44 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 9 y^{6}-27 y^{4}+18 y^{2}+6 $$

Step-by-Step Solution

Answer

Factor out 3 to get \(3(y^6 - 9y^4 + 6y^2 + 2)\).

Step 1: Identify the Greatest Common Factor (GCF)

To factor the polynomial, we need to first determine the greatest common factor (GCF) of its terms. The polynomial is composed of the terms: \(9y^6\), \(-27y^4\), \(18y^2\), and \(6\). Identify the GCF of the coefficients (9, 27, 18, 6) and the GCF of the variable powers (if applicable). The GCF of the coefficients is 3, and since not all terms have the variable \(y\), we only extract the numerical GCF, which is 3.

Step 2: Factor Out the GCF

Once the GCF is identified, factor it out from each term in the polynomial. This means dividing each term by 3: \[9y^6 \div 3 = 3y^6\] \[-27y^4 \div 3 = -9y^4\]\[18y^2 \div 3 = 6y^2\]\[6 \div 3 = 2\]Thus, when the GCF (3) is factored out, the polynomial becomes:\[3(y^6 - 9y^4 + 6y^2 + 2)\]

Step 3: Verify the Factored Expression

Double-check that the factored expression is correct by distributing the GCF back into the factored polynomial, which should return to the original expression:Distribute 3:\[3(y^6 - 9y^4 + 6y^2 + 2) = 3y^6 - 27y^4 + 18y^2 + 6\] This matches the original polynomial, confirming that the factoring is correct.

Key Concepts

Greatest Common Factor

The Greatest Common Factor, or GCF, is a key concept in factoring polynomials. It refers to the largest number that can divide all terms in an expression evenly. For instance, when factoring polynomials like \(9y^6 - 27y^4 + 18y^2 + 6\), we look at each term's coefficients: 9, 27, 18, and 6. By breaking these numbers down into their prime factors—you'll notice they share a common factor of 3.

To find the GCF:

• Prime factorize each number in the expression's coefficients.
• Identify the common prime factors.
• The GCF of the values is the product of the smallest powers of shared factors.

This method ensures that we find the correct GCF for numbers in a polynomial, which can greatly simplify the expression when factoring it.

Algebra

Algebra is an essential branch of mathematics that deals with symbols and the rules for manipulating those symbols. In the context of factoring polynomials, algebra helps us break down complicated expressions. Polynomial expressions are algebraic, meaning they contain variables and coefficients.

When working with algebra, you should remember:

• Variables represent unknown values and can be manipulated as numbers.
• Coefficients are the numerical part of terms involving variables.
• Operations such as addition, subtraction, multiplication, and division can be performed on these terms.

Using algebra, we can reorganize and simplify polynomial expressions, making them easier to interpret and solve, especially when factoring them by extracting the GCF.

Polynomial Expressions

Polynomial expressions are sums or differences of terms with variables raised to whole number powers. These terms are algebraic expressions involving a variable (often denoted as \(x\) or \(y\)) and exponents. In the polynomial \(9y^6 - 27y^4 + 18y^2 + 6\), you see terms like \(9y^6\) and \(-27y^4\).

The characteristics of polynomial expressions include:

• Each term consists of a coefficient and a variable part.
• Exponents of all terms are whole numbers.
• They can be simple, like \(x^2 + 3\), or more complex, like the polynomial above.

This understanding is crucial when factoring, as it ensures you know how each part contributes to the expression's overall structure. Recognizing patterns allows for easier factoring, like identifying a common factor in polynomial expressions.

Variable Powers

Variable powers within polynomials are the exponents applied to variables. They indicate how many times the variable is used as a factor. Understanding how to work with variable powers is vital in factoring because it influences the factorization process.

Key points include:

• The power of a variable indicates repeated multiplication.
• When factoring, focus on the highest power first.
• In \(9y^6 - 27y^4 + 18y^2 + 6\), notice the powers: 6, 4, and 2.

If every term in the polynomial includes the same variable, factor out the lowest power of that variable from each term. However, if one or more terms lack the variable, only factor out the GCF from the coefficients. This approach streamlines solving polynomial equations efficiently by properly handling variable powers.