Problem 78 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. Then factor by grouping. $$ 90+15 y^{2}-18 x-3 x y^{2} $$

Step-by-Step Solution

Answer

The factored form is \(3(5-x)(6+y^2)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, we need to find the greatest common factor of all terms in the polynomial: \[ 90 + 15y^2 - 18x - 3xy^2 \]The coefficients are 90, 15, 18, and 3. The GCF of these numbers is 3. There are no common variables among all terms, so the GCF is 3.

Step 2: Factor Out the GCF

Divide each term in the polynomial by the GCF (which is 3) and rewrite the polynomial in factored form:\[ 3(30 + 5y^2 - 6x - xy^2) \]

Step 3: Group the Terms

Now, focus on the expression inside the parentheses. Group the terms to make it easier to factor further:\[ 3((30 - 6x) + (5y^2 - xy^2)) \]

Step 4: Factor Each Group

Factor out the common factor from each group:- From \(30 - 6x\), factor out 6: \[ 6(5 - x) \] - From \(5y^2 - xy^2\), factor out \(y^2\): \[ y^2(5 - x) \] Thus, we rewrite:\[ 3(6(5 - x) + y^2(5 - x)) \]

Step 5: Factor by Grouping

Notice that \(5 - x\) is a common factor in both groups. Factor \(5 - x\) out of the expression:\[ 3((5 - x)(6 + y^2)) \]

Key Concepts

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is a fundamental concept in algebra that helps simplify polynomial expressions. It refers to the largest number that divides all the coefficients of the terms in an expression without a remainder. In our example, the polynomial is \[90 + 15y^2 - 18x - 3xy^2.\] When determining the GCF of this polynomial, we focus on the numbers: 90, 15, 18, and 3.

• The GCF of these coefficients is 3.
• Since there are no variables common in all terms, only the numerical GCF is considered.

Finding the GCF is a helpful first step, as it allows us to factor it out and simplify the polynomial into a form that is easier to handle further.

Factoring by Grouping

Factoring by grouping is a technique used when a polynomial seems too complex or does not fit into simpler factoring scenarios. After factoring out the GCF in our problem, we obtain: \[3(30 + 5y^2 - 6x - xy^2).\]Here is where grouping becomes essential:

• We form two groups from the terms in the polynomial; for example, \[(30 - 6x)\]and \[(5y^2 - xy^2).\]
• Each group is then factored independently. This includes finding any common factor within each group.
• For \[(30 - 6x),\] we factor out 6, and for \[(5y^2 - xy^2),\] we factor out \[y^2.\]

Ultimately, grouping helps to transform the polynomial into expressions where a common factor in all groups can be further factored out, significantly simplifying the equation.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. They form the backbone of algebra.

• Polynomials are characterized by terms like \[15y^2, -18x,\] and \[-3xy^2,\] each separated by a plus or minus sign.
• Each term can have constants (numbers on their own), variables (like \[x\] or \[y\]), or a product of these.
• The degree of a polynomial is determined by the highest power of the variable in any term.

Understanding the structure of polynomials is important for applying various algebraic techniques like factoring by grouping or distributing terms correctly.

Algebraic Techniques

Algebraic techniques are methods used to manipulate or simplify algebraic expressions. Factoring, as we applied in this problem, is just one of the many techniques.

• Finding the GCF is crucial for simplifying expressions at the start.
• Grouping allows for breaking down complex expressions into more manageable parts.
• Recognizing patterns, such as common factors or similar terms, aids in effective problem-solving.
  These techniques are not only useful for solving equations but also for understanding and verifying polynomial identities, simplifying rational expressions, and even solving real-world problems.

Proficiency in algebraic techniques strengthens one's ability to work with mathematical models and fosters a deeper comprehension of algebra's role in calculus and beyond.