Problem 133 - Elementary Algebra

Question

Will factored \(x^{2}-x-20\) as \((x+5)(x-4)\). Bill factored it as \((x+4)(x-5)\). Phil factored it as \((x-5)(x-4) .\) Who is correct? Explain why the other two are wrong.

Step-by-Step Solution

Answer

Bill is correct. His factorization \((x+4)(x-5)\) simplifies to the original polynomial \(x^2 - x - 20\).

Step 1: Identify the Polynomial

The given polynomial is: \[x^2 - x - 20\]We are to determine if any of the factorizations provided by Will, Bill, and Phil are correct.

Step 2: Verify Will's Factorization

Will's factorization is \((x+5)(x-4)\). To verify this, we can expand it back to the original polynomial:\[(x+5)(x-4) = x(x-4) + 5(x-4)\]\[= x^2 - 4x + 5x - 20\]\[= x^2 + x - 20\]Will's factorization does not simplify to \(x^2 - x - 20\) but to \(x^2 + x - 20\), making it incorrect.

Step 3: Verify Bill's Factorization

Bill's factorization is \((x+4)(x-5)\). Again, expand it back to the polynomial:\[(x+4)(x-5) = x(x-5) + 4(x-5)\]\[= x^2 - 5x + 4x - 20\]\[= x^2 - x - 20\]Bill's factorization simplifies correctly to \(x^2 - x - 20\).

Step 4: Verify Phil's Factorization

Phil's factorization is \((x-5)(x-4)\). To check, expand it back:\[(x-5)(x-4) = x(x-4) - 5(x-4)\]\[= x^2 - 4x - 5x + 20\]\[= x^2 - 9x + 20\]Phil's factorization simplifies to \(x^2 - 9x + 20\), which is not the original polynomial making it incorrect.

Step 5: Conclusion

Bill's factorization \((x+4)(x-5)\) is the only one that correctly simplifies to the original polynomial \(x^2 - x - 20\). Will and Phil both used incorrect factorizations.

Key Concepts

quadratic equations

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). It is called quadratic because it involves the square term \(x^2\). Solving quadratic equations is fundamental in algebra and often requires factoring the polynomial.
For example, consider the equation given in the exercise: \(x^2 - x - 20 = 0\). Here, \(a = 1\), \(b = -1\), and \(c = -20\). The goal is to factor this equation into the product of two binomials.

polynomial expansion

Polynomial expansion involves multiplying binomials and combining like terms. This technique helps verify if a given factorization is correct.
For instance, to verify Will's factorization \((x+5)(x-4)\):

\t
• Multiply the binomials: \t\( (x+5)(x-4) = x^2 - 4x + 5x - 20 \)
\t
• Combine like terms: \t\( x^2 + x - 20 \).

This result does not match the original polynomial \(x^2 - x - 20\), thus proving Will's solution incorrect.
Similarly, to verify Bill's factorization \((x+4)(x-5)\):

\t
• Multiply the binomials: \( (x+4)(x-5) = x^2 - 5x + 4x - 20 \).
\t
• Combine like terms: \( x^2 - x - 20 \).

This matches the original polynomial, making Bill's factorization correct.

algebraic verification

Algebraic verification ensures the accuracy of factorization by expanding the factors back into the original form.
To verify Phil's factorization \((x-5)(x-4)\):

\t
• Multiply the binomials: \t\( (x-5)(x-4) = x^2 - 4x - 5x + 20 \)
\t
• Combine like terms: \( x^2 - 9x + 20 \).

This does not match the original equation \(x^2 - x - 20\), indicating Phil's solution is incorrect. Through algebraic verification, we confirm Bill's factorization is right because expanding \( (x+4)(x-5) \) brings us back to the original polynomial.

factorization techniques

Factorization techniques are methods used to express a polynomial as the product of its factors. These techniques are crucial for simplifying and solving quadratic equations.
For the polynomial \(x^2 - x - 20\), we look for two numbers that multiply to \(-20\) (the constant term) and add up to \(-1\) (the coefficient of x).
In this case, the numbers are \(4\) and \(-5\).
Hence, the correct factorization is \( (x+4)(x-5) \).

\t
• Identify a and c from the polynomial \( ax^2 + bx + c \).
\t
• Find two numbers that multiply to get ac (in this case, -20) and add to get b (in this case, -1).
\t
• Split the middle term using these two numbers.
\t
• Factor by grouping to find the final factors.

Therefore, \( (x+4)(x-5) \) is the correct factorization technique applied, as was accurately concluded for Bill's factorization.