Problem 38 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$y^{2}-18 y-81$$

Step-by-Step Solution

Answer

The factorized form of the given quadratic equation \(y^{2}-18y-81\) is \((y - 9)(y + 9)\).

Step 1: Identify the coefficients

In the given quadratic equation \(y^{2}-18y-81\), the coefficient of \(y^{2}\) (a) is 1, the coefficient of y (b) is -18, and the constant term (c) is -81.

Step 2: Find the factors

We need to find two numbers that multiply to -81 and add up to -18. The numbers are -9 and 9, because (-9) * 9 equals -81 and (-9) + 9 equals 0.

Step 3: Write the factorized equation

Replace the coefficients in the form \((variable - factor1)(variable - factor2)\). In this case, the factorized quadratic equation would be \((y - 9)(y + 9)\).

Step 4: Check the factors

Multiply the factors back together to verify the original equation. (\(y - 9) * (y + 9)\) simplifies back to \(y^{2}- 18y - 81\), proving the factorization to be correct.

Key Concepts

Quadratic Equation Solving

Understanding how to solve quadratic equations is fundamental in algebra. A quadratic equation typically has the form \( ax^2 + bx + c = 0 \) where \( a \) is the coefficient of the squared term, \( b \) is the linear term's coefficient, and \( c \) is the constant term.

To solve, we often search for two numbers that satisfy specific conditions related to the coefficients in the equation. These conditions usually involve the numbers adding to equal \( b \) and multiplying to be \( ac \) (if the leading coefficient \( a \) is not 1). With these numbers, we can factor the equation or apply the quadratic formula \(-\frac{b \pm \sqrt{b^2-4ac}}{2a}\) for finding the solutions. Factorization is preferable when it's possible because it often simplifies the process.

Identifying Coefficients

To work with quadratic equations effectively, first, we must identify the coefficients correctly. Coefficients are the numerical or constant parts of the terms in a polynomial. In the quadratic term \( ax^2 \) the coefficient is \( a \) and it is integral in determining the shape of the graph. For the linear term \( bx \) in the equation, \( b \) affects the slope, and the constant term \( c \) is where the graph intersects the y-axis when \( x = 0 \).

When we know these coefficients, we can easily plot the graph of the quadratic, use them to factor the equation or find the vertex of the parabola. Clear identification is crucial to avoid mistakes in subsequent calculations.

Finding Factors of Polynomials

Factoring polynomials requires finding an expression that consists of two or more simpler expressions multiplied together to produce the original polynomial. For a quadratic, we often look for two binomial factors that can be valid when multiplied together.

Factorization Techniques:

• Guess and Check: Using knowledge of numbers, we guess pairs that multiply to the constant term and add to the coefficient of \( x \).
• Factoring by grouping: In some cases, we can group terms and factor common elements out to find pairs of binomials.
• Using formulae: Some quadratics can be factored by applying special product formulae such as the difference of squares or perfect square trinomials.

Understanding these methods and how to apply them is vital when tackling more complex polynomials.

Verification of Factorization

After factoring a polynomial, it's essential to verify that we've done it correctly. We do this by multiplying the factors back together to check if we get the original polynomial.

If we correctly factor \( y^2 - 18y - 81 \) into \( (y - 9)(y + 9) \), re-multiplying the factors should yield the original expression. If we don't, we might need to revisit the factorization process and correct any mistakes. Verification builds our confidence that factorization is accurate, ensuring our understanding and work's integrity. Moreover, it's a step towards ensuring that any subsequent work based on that factorization is also correct.