Problem 19 - A Graphical Approach to College Algebra

Question

Layla factored \(16 a^{2}-40 a-6 a+15\) by grouping and got an answer of \((8 a-3)(2 a-5) .\) Jamal factored the same polynomial and obtained \((3-8 a)(5-2 a) .\) Are both of these answers correct? If not, why not?

Step-by-Step Solution

Answer

Both answers are correct; they represent the same polynomial differently.

Step 1: Verify Layla's Factorization

We start by expanding Layla's factorization to see if it matches the original polynomial. Layla factored the polynomial as \((8a - 3)(2a - 5)\). Let's expand this product:\[(8a - 3)(2a - 5) = 8a \cdot 2a + 8a \cdot (-5) + (-3) \cdot 2a + (-3) \cdot (-5)\]This simplifies to:\[16a^2 - 40a - 6a + 15\]After combining like terms, we have:\[16a^2 - 46a + 15\]This matches the original polynomial, so Layla's factorization is correct.

Step 2: Verify Jamal's Factorization

Now we expand Jamal's factorization. Jamal factored the polynomial as \((3 - 8a)(5 - 2a)\). Let's expand this product:\[(3 - 8a)(5 - 2a) = 3 \cdot 5 + 3 \cdot (-2a) + (-8a) \cdot 5 + (-8a) \cdot (-2a)\]This simplifies to:\[15 - 6a - 40a + 16a^2\]Combine the like terms to get:\[16a^2 - 46a + 15\]Since this expanded expression matches the original polynomial, Jamal's factorization is also correct.

Step 3: Compare Factored Forms

Both expressions \((8a - 3)(2a - 5)\) and \((3 - 8a)(5 - 2a)\) expand to the polynomial \(16a^2 - 46a + 15\). We observe that Jamal's expression is equivalent to Layla's, just represented differently. Notice that by factoring a \(-1\) from both terms in \((3 - 8a)\) and \((5 - 2a)\), or rewriting them as numeric opposites, you can modify the terms from one factored form to the other.

Key Concepts

Polynomial Expansion

Polynomial expansion involves multiplying terms in polynomial expressions to write them as a sum or difference of terms.
Each term in the first polynomial is multiplied by each term in the second polynomial.
For instance, when expanding \((8a - 3)(2a - 5)\), we perform the following steps:

• Multiply the first terms: \(8a \cdot 2a = 16a^2\)
• Multiply the outer terms: \(8a \cdot (-5) = -40a\)
• Multiply the inner terms: \((-3) \cdot 2a = -6a\)
• Multiply the last terms: \((-3) \cdot (-5) = 15\)

Then, we combine the resulting terms to get: \[16a^2 - 40a - 6a + 15,\] which simplifies to \(16a^2 - 46a + 15\).
This matches the original polynomial, ensuring the factorization was done correctly.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms.
It's especially useful when there's no common factor amongst all terms.
Here’s how it works:

• Group the terms in pairs: \(16a^2 - 40a - 6a + 15\) can be grouped as \((16a^2 - 40a)\) and \((-6a + 15)\).
• Factor out the greatest common factor from each group: From \((16a^2 - 40a)\), we factor out \(8a\), and from \((-6a + 15)\), we factor out \(-3\).
• This gives us: \(8a(2a - 5) - 3(2a - 5)\).
• Notice that \((2a - 5)\) is a common factor: We can now factor \((2a - 5)\) out to get \((8a - 3)(2a - 5)\).

This method simplifies complex expressions and verifies whether two different forms are equivalent.

Polynomial Equivalence

Polynomial equivalence means two expressions are equal, even if they look different.
This is crucial when comparing different factored forms of the same polynomial.
For example, Layla’s \((8a - 3)(2a - 5)\) and Jamal’s \((3 - 8a)(5 - 2a)\) both expand to \(16a^2 - 46a + 15\).

• To confirm equivalence, expand both expressions back to the standard polynomial form.
• If they produce identical results, they are equivalent, as seen in this scenario.
• Rewriting one form into another can involve factoring out constants or rearranging terms.

This shows that different factoring paths can lead to the same solution, highlighting the flexibility and power of polynomials.