Problem 30 - Algebra and Trigonometry

Question

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{3}+3 x^{2}-4 x-12 $$

Step-by-Step Solution

Answer

The factored form is \((x + 3)(x - 2)(x + 2)\), with zeros at \(-3, 2, -2\).

Step 1: Look for Common Factors

Check if there is a greatest common factor (GCF) for all the terms. In this polynomial, the terms don't share any common factor other than 1, so we can skip factoring out any GCF.

Step 2: Group Terms for Factoring by Grouping

We can attempt to factor by grouping. Group the terms: \[ P(x) = (x^3 + 3x^2) + (-4x - 12) \] This groups the polynomial into two pairs of terms.

Step 3: Factor Each Group

Factor out the greatest common factor from each group: \[ P(x) = x^2(x + 3) - 4(x + 3) \] Identify the common binomial \((x + 3)\) in both groups.

Step 4: Factor the Entire Expression

Since \((x + 3)\) is common, factor it out: \[ P(x) = (x + 3)(x^2 - 4) \] You now have a product of two factors.

Step 5: Recognize Further Factorization

Notice that \(x^2 - 4\) is a difference of squares, which can be factored as: \[ x^2 - 4 = (x - 2)(x + 2) \] So, the expression becomes: \[ P(x) = (x + 3)(x - 2)(x + 2) \]

Step 6: Find the Zeros of the Polynomial

Set each factor equal to zero to find the zeros: 1. \(x + 3 = 0 \Rightarrow x = -3\)2. \(x - 2 = 0 \Rightarrow x = 2\)3. \(x + 2 = 0 \Rightarrow x = -2\)Thus, the zeros are \(x = -3, 2, -2\).

Step 7: Sketch the Graph

Knowing the zeros \(x = -3, 2, -2\) helps sketch the curve. Start at negative infinity on a cubic curve, crossing the x-axis at these points. Since it's a cubic polynomial with a positive leading coefficient, it extends to positive infinity as \(x\) goes to positive infinity.

Key Concepts

Finding Zeros

Finding the zeros of a polynomial is a crucial task in algebra. Zeros are the x-values where the polynomial evaluates to zero. Simply put, they're the points where the graph of the polynomial touches or crosses the x-axis.

To find the zeros, first, factor the polynomial completely. Once in factored form, set each factor equal to zero. For instance, in the given polynomial \(P(x) = (x + 3)(x - 2)(x + 2)\), we find the zeros by solving the following equations:

• \(x + 3 = 0\Rightarrow x = -3\)
• \(x - 2 = 0\Rightarrow x = 2\)
• \(x + 2 = 0\Rightarrow x = -2\)

Hence, the zeros are \(x = -3, -2, 2\). These zeros tell us the x-intercepts of the graph, and knowing them is helpful for graph sketching and understanding the behavior of the polynomial function.

Difference of Squares

A difference of squares is a specific form that can be easily factored. It appears as \(a^2 - b^2\) and always factors into the product \((a - b)(a + b)\).

In the polynomial \(P(x) = x^2 - 4\), we notice that this fits the form of a difference of squares. Here, \(a = x\) and \(b = 2\). By applying the formula, we factor \(x^2 - 4\) into \((x - 2)(x + 2)\). Recognizing these forms simplifies the factoring process significantly.

Understanding and identifying difference of squares quickly can save time and reduce complexity when factoring.

Graph Sketching

Sketching a graph from the factored form of a polynomial gives a visual insight into the equation. Knowing the zeros \(x = -3, -2, 2\) provides critical points where the polynomial intersects the x-axis.

For a cubic polynomial like \(P(x) = (x + 3)(x - 2)(x + 2)\), the graph’s general shape will start from negative infinity, cross the x-axis at the zeros, and shoot towards positive infinity because the leading term \(x^3\) is positive.

Additionally, knowing it is cubic implies that on a large scale, the graph will resemble an "S" shape. Use the zeros to determine sections where the graph rises and falls. Summary sketching using these intercepts and understanding the end behavior of polynomials helps in effectively drawing the graph.

Factoring by Grouping

Factoring by grouping is a technique often used when dealing with polynomials that initially have more than two terms. It involves grouping terms to find common factors within those groups.

In the exercise, the polynomial \(P(x) = x^3 + 3x^2 - 4x - 12\) is grouped as \((x^3 + 3x^2) + (-4x - 12)\). Within each group, we find the common factors:

• Factor \(x^2\) from \(x^3 + 3x^2\), giving \(x^2(x + 3)\)
• Factor \(-4\) from \(-4x - 12\), giving \(-4(x + 3)\)

Notice \((x + 3)\) is common, so it can be factored out: \((x + 3)(x^2 - 4)\). Grouping helps break down complex polynomials into simpler, solvable parts and is a valuable tool in solving and factoring polynomials efficiently.