Problem 33 - Algebra and Trigonometry

Question

\(27-40\) Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{3}+x^{2}-x-1 $$

Step-by-Step Solution

Answer

The zeros are \(x = -1\) (multiplicity 2) and \(x = 1\); graph the cubic with these crossings.

Step 1: Group and Factor by Grouping

The polynomial is given as \( P(x) = x^3 + x^2 - x - 1 \). Start by grouping terms: \( (x^3 + x^2) + (-x - 1) \). Factor out the common factors in each group: \( x^2(x + 1) - 1(x + 1) \).

Step 2: Factor Out Common Binomial

Notice the common binomial \((x+1)\) in each grouped term. Factor it out: \( (x + 1)(x^2 - 1) \).

Step 3: Factor the Quadratic Polynomial

Now we factor the quadratic \(x^2 - 1\) using the difference of squares formula: \( x^2 - 1 = (x + 1)(x - 1) \). Thus the polynomial becomes \( (x + 1)^2(x - 1) \).

Step 4: Find the Zeros

The factored form \( (x + 1)^2(x - 1) \) indicates the zeros are when \( x + 1 = 0 \) or \( x - 1 = 0 \). Solving these equations gives zeroes at \( x = -1 \) with multiplicity 2, and \( x = 1 \).

Step 5: Sketch the Graph

Plot the zeros on the x-axis: a double root at \( x = -1 \) (meaning the graph touches but doesn't cross the x-axis) and a single root at \( x = 1 \). Since the leading term \( x^3 \) is positive, the graph starts in the negative quadrant and ends in the positive quadrant, creating a cubic shape characteristic of such polynomials.

Key Concepts

Factoring by Grouping

Factoring by grouping is a strategic method for breaking down polynomials into simpler factors. It's especially useful when a polynomial does not easily factor into known patterns like perfect squares or cubes. In this method, you group the terms in a way that allows you to factor out a common expression. Consider the polynomial given in our exercise: \[ P(x) = x^3 + x^2 - x - 1 \] The idea is to rearrange and group the terms. Here, group the first two terms together and the last two terms together:\[ (x^3 + x^2) + (-x - 1) \] For each group, we factor out the greatest common factor. For the first group, \(x^2\) is the common factor, leaving us with \(x^2(x + 1)\). In the second group, factor out a \(-1\), resulting in \[ (x^3 + x^2) + (-x - 1) = x^2(x + 1) - 1(x + 1) \] This method is particularly effective when a common binomial, such as \((x + 1)\), emerges among the grouped terms.

Difference of Squares

The difference of squares is a valuable formula that assists in factoring certain quadratic expressions. It exploits the identity: \[ a^2 - b^2 = (a + b)(a - b) \] which highlights that any expression of this form can be separated into two binomial factors. In our current polynomial, after factoring by grouping, we are left with the expression \( x^2 - 1 \). Here, recognize it as a difference of squares because it can be rewritten as \[ x^2 - 1^2 \] Using the difference of squares formula, it factors further into\[ (x + 1)(x - 1) \] Some quadratics or polynomials inherently fit this form, allowing us to factor them quickly and easily with this formula. It's a handy tool when dealing with polynomials where the terms are perfect squares.

Finding Zeros

Finding zeros of a polynomial is crucial, as these values indicate where the polynomial graph intersects or touches the x-axis. With the factored form of \[ P(x) = (x + 1)^2(x - 1) \] we set each factor to zero to find the roots. Solving the equations:

• \( (x + 1)^2 = 0 \) implies \( x = -1 \), and with multiplicity 2, this means the graph only touches but doesn’t cross the x-axis.
• \( x - 1 = 0 \) implies \( x = 1 \).

Therefore, the zeros are at \(x = -1\) and \(x = 1\). The zero at \(x = -1\) suggests a unique behavior because of its multiplicity, meaning a sort of flatness or a touch point there, rather than a crossing.

Polynomial Graph Sketching

Graphing polynomials offers a visual representation of their behavior across different values of \(x\). Start by plotting the zeros found, as they indicate critical points where the polynomial intersects the horizontal x-axis. For \[ P(x) = (x + 1)^2(x - 1) \] mark points at \(x = -1\) and \(x = 1\) on the x-axis. Remember, \(x = -1\) has a multiplicity of 2, so the graph only touches the axis here. At \(x = 1\), it crosses.

• Since the highest power of \(x\) is 3 (from the original \(x^3\)), the graph is cubic.
• The leading coefficient is positive, meaning the graph begins in the lower left quadrant (where y is negative) and rises to the upper right quadrant.

The general shape is an 'S' curve. To complete the sketch, consider also drawing a few more points around the zeros to see how it behaves between them. This way, you'll have a confident idea of its overall path.