Problem 59 - Algebra and Trigonometry

Question

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=2 x^{3}-7 x^{2}+4 x+4 $$

Step-by-Step Solution

Answer

Zeros are \(x = -\frac{1}{2}\) and \(x = 2\) (double root). Graph shows end behavior from \(-\infty\) to \(\infty\).

Step 1: Analyze the Polynomial Degree

The given polynomial function, \(P(x) = 2x^3 - 7x^2 + 4x + 4\), is a cubic polynomial because the highest power of \(x\) is 3. A cubic polynomial can have up to 3 real roots.

Step 2: Determine the Possible Rational Zeros

By the Rational Root Theorem, any rational zero, \( \frac{p}{q} \), of \( P(x) \) must be a factor of the constant term (4) divided by a factor of the leading coefficient (2). The possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2}\).

Step 3: Test Each Possible Rational Zero

Evaluate \(P(x)\) at each possible rational zero. Check \(P(1)\), \(P(-1)\), \(P(2)\), etc., to see if any of these values make the polynomial equal to zero. Upon testing:- \(P(1) = 2(1)^3 - 7(1)^2 + 4(1) + 4 = 3\) (not zero)- \(P(-1) = 2(-1)^3 - 7(-1)^2 + 4(-1) + 4 = -9\) (not zero)- \(P(2) = 2(2)^3 - 7(2)^2 + 4(2) + 4 = 0\) (zero, so \(x=2\) is a root)Now divide the original polynomial by \(x-2\).

Step 4: Perform Synthetic Division

Perform synthetic division of \(P(x)\) by \(x - 2\):Dividing yields \(2x^2 - 3x - 2\). The quotient is a quadratic polynomial.

Step 5: Factor the Quotient Polynomial

Factor the quadratic polynomial \(2x^2 - 3x - 2\). This can be written as \((2x + 1)(x - 2)\), giving the complete factorization of \(P(x)\) as \(P(x) = (x - 2)(2x^2 - 3x - 2)\).

Step 6: Solve for Remaining Zeros

Set each factor to zero to find the remaining zeros:- For \(2x + 1 = 0\), solve to get \(x = -\frac{1}{2}\).- \(x = 2\) was found earlier but appears as a repeated root.

Step 7: Sketch the Graph Using Zeros and End Behavior

Sketch the graph of \(P(x)\):- Zeros at \(x = -\frac{1}{2}, 2\) (2 as a double root).- End behavior: As \(x \to \infty\), \(P(x) \to \infty\), and as \(x \to -\infty\), \(P(x) \to -\infty\).- The graph crosses the x-axis at \(x = -\frac{1}{2}\) and touches the x-axis at \(x=2\) without crossing due to the double root.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool that helps us predict the possible rational zeros of a polynomial.
When faced with a polynomial, it can be daunting to guess what inputs (roots) will make the polynomial equal to zero.
The Rational Root Theorem simplifies this by offering a list of potential candidates. Here's how it works:

• First, identify the constant term and the leading coefficient of the polynomial. In the polynomial \( P(x) = 2x^3 - 7x^2 + 4x + 4 \), the constant term is \( 4 \) and the leading coefficient is \( 2 \).
• According to the theorem, any rational zero \( \frac{p}{q} \) must have \( p \), a factor of the constant term \( 4 \), and \( q \), a factor of the leading coefficient \( 2 \).
• The factors of \( 4 \) are \( \pm 1, \pm 2, \pm 4 \) and the factors of \( 2 \) are \( \pm 1, \pm 2 \).
• This provides a list of possible rational roots: \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2} \).

By calculating and evaluating these possibilities, one can quickly zero in on the actual roots, if they are rational.

cubic polynomial

A cubic polynomial is a type of polynomial of degree three.
It takes the general form \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \).
In our example, \( P(x) = 2x^3 - 7x^2 + 4x + 4 \), clearly fits this form.Key characteristics of cubic polynomials include:

• They can have up to three real roots (solutions) because it's a third-order polynomial.
• The graph of a cubic polynomial typically has a characteristic "S" shape, representing its three degrees and inflection point.
• As \( x \to \infty \), a cubic function \( ax^3 \) tends towards \( \, ext{either}\, \pm \infty \) depending on the leading coefficient \( a \).

Understanding the basic properties of cubic polynomials aids in graph interpretation and root analysis through algebraic methods, like factoring and division.

synthetic division

Synthetic division is a short method for dividing a polynomial by a linear binomial of the form \( x - r \).
It's especially useful when applying the Rational Root Theorem to test potential roots.
Here's a brief overview of how synthetic division works:

• First, write down the coefficients of the polynomial. For \( P(x) = 2x^3 - 7x^2 + 4x + 4 \), write the coefficients: \( 2, -7, 4, 4 \).
• Then, write the value of \( r \) (the test root) off to the left. Let's take \( r = 2 \) as found in our example.
• Perform synthetic division by bringing down the leading coefficient and repeatedly multiplying and adding.
• This division process produces a new polynomial (the quotient) and determines if \( x - r \) is a factor by checking the remainder.

In our case, dividing by \( x-2 \) results in \( 2x^2 - 3x - 2 \) with no remainder, proving that \( x = 2 \) is a root.
This efficient process makes it easier to factor polynomials and simplify root finding.