Which of the following is/are the factor(s) of \((3x + y)^2 + (3x+y)(x+5y)-20(x+5y)^2\)? I. \((4x+13y)\) II. \((x + 19y)\) Select the correct answer using the code given below.
Factoring the Algebraic Expression
We are asked to find the factors of the expression: \((3x + y)^2 + (3x+y)(x+5y)-20(x+5y)^2\). Let's simplify this expression by using substitution.
Using Substitution for Simplification
To make the expression easier to handle, let's substitute:
- Let \(A = (3x + y)\)
- Let \(B = (x+5y)\)
Substituting these into the original expression gives us:
\(A^2 + AB - 20B^2\)
Factoring the Quadratic Form
The expression \(A^2 + AB - 20B^2\) is a quadratic expression in terms of \(A\) and \(B\). We can factor it similar to how we factor a standard quadratic equation like \(z^2 + z - 20\). We need two numbers that multiply to \(-20\) and add up to \(1\). These numbers are \(5\) and \(-4\).
Therefore, we can factor the expression as:
\((A + 5B)(A - 4B)\)
Substituting Back to Original Terms
Now, let's substitute back \(A = (3x + y)\) and \(B = (x+5y)\) into the factored form:
First Factor: \((A + 5B)\)
Substitute the values of A and B:
\((3x + y) + 5(x+5y)\)
Distribute the \(5\):
\(3x + y + 5x + 25y\)
Combine like terms:
\(8x + 26y\)
We can factor out a \(2\) from this expression:
\(2(4x + 13y)\)
This shows that \((4x + 13y)\) is a factor of the original expression.
Second Factor: \((A - 4B)\)
Substitute the values of A and B:
\((3x + y) - 4(x+5y)\)
Distribute the \(-4\):
\(3x + y - 4x - 20y\)
Combine like terms:
\(-x - 19y\)
We can factor out a \(-1\) from this expression:
\(-(x + 19y)\)
This shows that \((x + 19y)\) is a factor of the original expression.
Conclusion on Factors
The original expression factors completely into \(2(4x + 13y) \times -(x + 19y)\), which simplifies to \(-2(4x + 13y)(x + 19y)\).
Based on our analysis:
- Factor I: \((4x+13y)\) is present in the factored form.
- Factor II: \((x+19y)\) is present in the factored form.
Therefore, both \((4x+13y)\) and \((x+19y)\) are factors of the given algebraic expression.
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