Problem 99 - Algebra A Combined Function

Question

The annual cotton crop yield (in 1000 bales) in the United States for the period \(2003-2007\) can be approximated by the polynomial \(-1264 x^{2}+5056 x+18,960,\) where \(x\) is the number of years after 2003. (Source: Based on data from the National Agricultural Statistics Service) a. Find the approximate amount of the cotton harvest in 2004. To do so, let \(x=1\) and evaluate \(-1264 x^{2}+5056 x+18,960\) b. Find the approximate amount of cotton harvested in 2007 . c. Factor the polynomial \(-1264 x^{2}+5056 x+18,960\).

Step-by-Step Solution

Answer

Cotton harvest in 2004 is 22,752 bales; in 2007 is 18,960 bales. Complex factorization yields roots indicating non-rational factors.

Step 1: Set up for 2004

To find the cotton harvest in 2004, we substitute \( x = 1 \) into the polynomial \(-1264x^2 + 5056x + 18960\).

Step 2: Solve for 2004 Harvest

Substituting \( x = 1 \), the expression becomes:\[-1264(1)^2 + 5056(1) + 18960\].Evaluate the terms: \(-1264 \cdot 1 = -1264\)\(5056 \cdot 1 = 5056\)Then:\[ -1264 + 5056 + 18960 = 22752 \]So, the approximate cotton harvest in 2004 is 22,752 bales.

Step 3: Setup for 2007

To find the cotton harvest in 2007, we substitute \( x = 4 \) into the polynomial \(-1264x^2 + 5056x + 18960\).

Step 4: Solve for 2007 Harvest

Substituting \( x = 4 \), the expression becomes:\[-1264(4)^2 + 5056(4) + 18960\].Evaluate the terms:\(4^2 = 16\)\(-1264 \cdot 16 = -20224\)\(5056 \cdot 4 = 20224\)Then:\[-20224 + 20224 + 18960 = 18960\]So, the approximate cotton harvest in 2007 is 18,960 bales.

Step 5: Setup for Factoring

To factor the polynomial \(-1264x^2 + 5056x + 18960\), look for common factors or use the quadratic equation method.

Step 6: Factoring the Polynomial

Notice all terms can be divided by 4:\[-4(316x^2 - 1264x - 4740)\]Attempt further factoring or apply the quadratic formula on \[316x^2 - 1264x - 4740\]. Finding roots and simplifying:After calculation, the factorization is complex, indicating non-rational factors, validation is required for an exact factorization over integers.

Key Concepts

Cotton Yield

The cotton yield in an agricultural context refers to the total production of cotton, often measured in bales, harvested over a specific period or from a set amount of land.
In this exercise, we're looking at the US cotton yield over several years, expressed through a polynomial function. This allows us to estimate how productive cotton fields were in that time frame.
The polynomial provided reflects a quadratic equation that models the yield in thousands of bales.To find the yield for a specific year, substitute the number of years after 2003 into the polynomial. The polynomial takes the form: \[-1264x^2 + 5056x + 18960\], where \(x\) is the difference in years from 2003.
For instance, to find the yield in 2004, we substitute \(x = 1\). This evaluates the function and gives an estimate of cotton production for that year.
Such calculated yields help in planning, policy-making, and predicting future crop success based on past patterns.

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into simpler components (factors) that, when multiplied, give the original polynomial.
This exercise requires factoring the quadratic polynomial \(-1264x^2 + 5056x + 18960\). Factoring polynomials can help simplify solving equations and understanding their roots.Here's a typical approach to factoring:

• Look for the greatest common factor (GCF). If all coefficients can be divided by a number, factor it out first.
• This polynomial can be simplified by noticing that each coefficient is divisible by 4, leading to: \(-4(316x^2 - 1264x - 4740)\).
• After this step, further factorization can be attempted on the trinomial \(316x^2 - 1264x - 4740\), but sometimes the factors aren’t neat integers.
• If factoring fails, using the quadratic formula provides solutions for roots, though they might not always be rational numbers.

Such methods show the versatility of polynomial expressions in algebra, where commuting complex expressions to simpler ones often helps in practical applications like calculating yield metrics.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, which generally have the form \(ax^2 + bx + c = 0\). In our exercise, this quadratic form is used to model cotton yield.
This structure is crucial because it shapes the curve representing real-world data, such as crop yields over time.What makes quadratic equations interesting is their characteristics:

• They form parabolas when graphed, depicting trends over time.
• The roots of the equation, which can be found via factoring, completing the square, or using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), indicate the \(x\)-values where the yield reaches zero.
• The vertex of the parabola represents the point of maximum or minimum yield depending on the parabola's direction, which is determined by the sign of \(a\).

In this context, because \(-1264x^2\) has a negative leading coefficient, the parabola opens downward, hinting at a peak yield somewhere before a decline, typically useful for studies in agricultural efficiency.