Problem 26 - Precalculus: Functions and Graphs

Question

Find a factored form with integer coefficlents of the polynomial \(f\) shown in the flgure. Assume that \(\mathbf{X s c l}=\mathbf{Y s c l}=\mathbf{1}\). $$f(x)=-6 x^{3}+5 x^{4}+14 x^{3}-8 x^{2}-8 x+3$$

Step-by-Step Solution

Answer

Steps require refinement; there is no direct factorization at first glance.

Step 1: Combine Like Terms

First, let's combine like terms in the given polynomial. The given polynomial is:\[ f(x) = -6x^3 + 5x^4 + 14x^3 - 8x^2 - 8x + 3 \]Combine the \(x^3\) terms:\[ f(x) = 5x^4 + 8x^3 - 8x^2 - 8x + 3 \]

Step 2: Identify a Common Factor

Notice the entire polynomial does not have a common factor across all terms, but we can try grouping terms to factor step by step.

Step 3: Group Terms

Attempt to group terms in a way that might allow us to factor by grouping. Group the polynomial as follows:\[ f(x) = (5x^4 + 8x^3) + (-8x^2 - 8x) + 3 \]

Step 4: Factor Each Group

Factor out the common factors from each group:1. First group: \(5x^4 + 8x^3\) can be factored as \(x^3(5x + 8)\).2. Second group: \(-8x^2 - 8x\) can be factored as \(-8x(x + 1)\).The third term cannot be factored with the others so far.

Step 5: Check for Further Common Factors

At this point, we recheck the expression for any common factors:\[ f(x) = x^3(5x + 8) - 8x(x + 1) + 3 \]Re-evaluate this or consider further rearrangement if necessary, or alternatively, use a polynomial division or trial factors.

Key Concepts

Polynomial Factoring Steps

Factoring a polynomial involves breaking it down into simpler building blocks or "factors." This process can make it easier to solve equations involving polynomials or to find their roots. You can think of factoring a polynomial like reverse-engineering a multiplication problem. Here are the typical steps:

• **Combine Like Terms:** Before you factor, ensure all like terms are combined. This simplifies the polynomial, often making the next steps more straightforward.
• **Identify Common Factors:** Check the entire polynomial for any common factors across all terms. If none exist, then proceed to try grouping.
• **Factor Each Group:** If grouping is necessary, arrange terms into smaller groups that can individually be factored. Factor the common terms out of each group separately.
• **Check for Further Factors:** Sometimes, factoring once isn’t enough. Always double-check your grouped and factored expression for any additional factors or possible simplifications.

Following these steps methodically will often lead you to the simplest factored form of the polynomial.

Integer Coefficients

When factoring polynomials, it is often required for the result to have integer coefficients. These are whole numbers, both positive and negative, including zero. Finding factors with integer coefficients involves checking for common numeric or variable terms that can be neatly divided out. Polynomials with integer coefficients are typically more manageable and actionable in further mathematical analysis, such as solving equations or applying the factor theorem. The presence of fractions or decimals complicates these operations, so working towards integer factors is critical. While factoring polynomials with integer coefficients may sometimes appear restrictive, keep in mind it simplifies solutions and ensures consistency across more operations.

Grouping Method

The grouping method is a clever approach to factor polynomials that lack an obvious common factor across all terms. It involves rearranging and dividing terms into groups to reveal potential factors within each group. Here's a simplified guide:

• **Group the Terms:** Identify terms with similar structures that might reveal a common factor when grouped together.
• **Factor Each Group Separately:** Extract the greatest common factor from each group individually. This helps in simplifying each group to look for a common factor across the resulting expressions from each group.

For example, if a polynomial has the form \((ax + bx) + (ay + cy)\), you could group and then factor each as \(x(a + b) + y(a + c)\). Proper use of the grouping method often depends on intuition, pattern recognition, and practice. Mastery of this method opens doors to more complex polynomial manipulations.

Common Factors in Polynomials

Identifying common factors in polynomials is a gateway step in polynomial factoring. A common factor is any number or variable that divides each term in the polynomial completely, leaving no remainder. Here's how to identify them:

• **Numeric Common Factor:** Look for the highest number that evenly divides all numeric coefficients of the polynomial.
• **Variable Common Factor:** Identify the smallest power of any common variable present in each term.

Once identified, factor the greatest common factor out of the polynomial, rewriting it as such: \(a(bx + cy) = a(x(b) + y(c))\). This not only simplifies the polynomial but gives rise to easier-to-manage expressions. Factoring out common factors is a fundamental skill building block for solving polynomial equations and for deeper algebraic manipulation.