Factor 112a^2c-112yk+392a^2k-32yc
The question is asking you to perform factorization on the given polynomial. Factorization is the process of breaking a complex expression down into a product of simpler expressions or factors. The polynomial provided is a linear combination of four terms: 112a^2c, -112yk, 392a^2k, and -32yc. The task is to find the common factors among these terms and express the polynomial as a product of these factors and the remaining terms. This could involve factoring out common numerical coefficients, variables, or combinations of both. The goal is to simplify the original expression into a product of polynomials that are easier to manage or analyze.
$112 a^{2} c - 112 y k + 392 a^{2} k - 32 y c$
Answer
Solution:
Step 1
Extract the common factor of $8$ from $112a^{2}c$ to get $8(14a^{2}c) - 112yk + 392a^{2}k - 32yc$.
Step 2
Remove the common factor of $8$ from $-112yk$ to obtain $8(14a^{2}c) + 8(-14yk) + 392a^{2}k - 32yc$.
Step 3
Take out the common factor of $8$ from $392a^{2}k$, resulting in $8(14a^{2}c) + 8(-14yk) + 8(49a^{2}k) - 32yc$.
Step 4
Pull out the common factor of $8$ from $-32yc$ to get $8(14a^{2}c) + 8(-14yk) + 8(49a^{2}k) + 8(-4yc)$.
Step 5
Combine the terms with the factor of $8$ to form $8(14a^{2}c - 14yk) + 8(49a^{2}k) + 8(-4yc)$.
Step 6
Merge the terms with the factor of $8$ to have $8(14a^{2}c - 14yk + 49a^{2}k) + 8(-4yc)$.
Step 7
Finally, factor out the $8$ from all terms to achieve $8(14a^{2}c - 14yk + 49a^{2}k - 4yc)$.
Knowledge Notes:
To factor an algebraic expression means to write it as a product of its factors. Factors are numbers or expressions that are multiplied together to get the original expression. The process of factoring involves:
Identifying Common Factors: Look for numbers, variables, or expressions that are common to each term in the expression.
Extracting Common Factors: Rewrite each term as a product of the common factor and another term. This step may involve division to find what remains after factoring out the common term.
Rewriting the Expression: Once the common factor is extracted from each term, the expression is rewritten as the product of the common factor and a new expression that includes the remaining terms.
Simplifying: If possible, further simplification may involve factoring the remaining expression or combining like terms.
In the given problem, the common factor of $8$ is identified and extracted from each term, and the expression is rewritten as the product of $8$ and another expression. This process is repeated until the entire original expression is factored. The use of parentheses is important to maintain the integrity of the terms as they are grouped together.
The use of LaTeX in this context is to clearly represent mathematical expressions and ensure that the numbers, variables, and operations are easily readable and understood. LaTeX is a typesetting system that is widely used for scientific documents, especially those that include mathematical content.
