Problem 41 - Algebra and Trigonometry

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=x^{3}+4 x^{2}+3 x-2 $$

Step-by-Step Solution

Answer

The real zeros are \( x = -2, \ x = -1 + \sqrt{2}, \ x = -1 - \sqrt{2} \).

Step 1: Check for Rational Zeros

To find potential rational zeros, apply the Rational Root Theorem. The possible rational zeros are the factors of the constant term divided by the factors of the leading coefficient. For the polynomial \( P(x) = x^3 + 4x^2 + 3x - 2 \), the factors of \(-2\) (the constant term) are \( \pm 1, \pm 2 \). Since the leading coefficient is 1, the possible rational zeros are \( \pm 1, \pm 2 \).

Step 2: Test Each Rational Zero

Substitute each possible rational zero into the polynomial: - For \( x = 1 \): \( P(1) = 1^3 + 4(1)^2 + 3(1) - 2 = 6 \) (not zero).- For \( x = -1 \): \( P(-1) = (-1)^3 + 4(-1)^2 + 3(-1) - 2 = -4 \) (not zero).- For \( x = 2 \): \( P(2) = 2^3 + 4(2)^2 + 3(2) - 2 = 24 \) (not zero).- For \( x = -2 \): \( P(-2) = (-2)^3 + 4(-2)^2 + 3(-2) - 2 = 0 \) (this is a zero of the polynomial).

Step 3: Perform Polynomial Division

Since \( x = -2 \) is a zero, divide the polynomial by \( x + 2 \) using synthetic division. The result of the division is:\[ x^3 + 4x^2 + 3x - 2 = (x + 2)(x^2 + 2x - 1) \]

Step 4: Find Zeros of Quadratic Factor

To find the zeros of \( x^2 + 2x - 1 \), use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1, b = 2, c = -1 \).Compute the discriminant: \( b^2 - 4ac = 2^2 - 4(1)(-1) = 4 + 4 = 8 \).Now, apply the formula: \[ x = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2} \].The zeros of this quadratic are \( x = -1 + \sqrt{2} \) and \( x = -1 - \sqrt{2} \).

Step 5: Compile All Zeros

The polynomial \( P(x) = x^3 + 4x^2 + 3x - 2 \) has the following real zeros: \( x = -2, \ x = -1 + \sqrt{2}, \ x = -1 - \sqrt{2} \). These include the rational zero and the zeros obtained from solving the quadratic.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is an essential tool when trying to find the rational zeros of a polynomial. It states that if there is a rational root, or zero, of a polynomial function, it can be expressed as a fraction \( p/q \), where:

• \( p \) is a factor of the constant term of the polynomial
• \( q \) is a factor of the leading coefficient

This is very useful because it narrows down the number of possible rational zeros you need to test. In our exercise, with the polynomial \( P(x) = x^3 + 4x^2 + 3x - 2 \), the constant term is \(-2\) and its factors are \( \pm 1 \) and \( \pm 2 \). The leading coefficient is \( 1 \), and its factors are \( \pm 1 \), making the possible rational zeros \( \pm 1 \) and \( \pm 2 \). By testing these potential zeros, we find which values satisfy the polynomial equation.

synthetic division

Synthetic division is a simpler method of dividing polynomials, and it's particularly handy when checking if a candidate from the Rational Root Theorem is indeed a zero. Once you have a suspected zero, you can use synthetic division to quickly divide the polynomial by \( x - r \), where \( r \) is your proposed zero.
This process involves:

• Taking the coefficients of the polynomial.
• Using the suspected zero to perform operations successively on these coefficients.
• Obtaining a new set of coefficients that form the quotient polynomial, with any remainder indicating whether or not the value is a true zero.

For example, in our exercise, since \( x = -2 \) was found to be a valid zero, synthetic division shows the polynomial factors into \( (x + 2)(x^2 + 2x - 1) \). This assists in breaking down complex polynomials into simpler quadratic forms for further solving.

quadratic formula

The Quadratic Formula is a powerful equation used to find the roots of a quadratic equation in the form \( ax^2 + bx + c = 0 \). It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula is versatile as it can handle any quadratic equation, regardless of whether it can be factored easily or not. In the context of our problem, after determining the polynomial can be factored into \( (x + 2)(x^2 + 2x - 1) \), we still need the quadratic formula to find the roots of the quadratic \( x^2 + 2x - 1 \). Thus, applying the quadratic formula gives us the roots \( x = -1 + \sqrt{2} \) and \( x = -1 - \sqrt{2} \).

discriminant

The discriminant is part of the quadratic formula that determines the nature and number of roots of a quadratic equation. It is the expression under the square root in the quadratic formula given by \( b^2 - 4ac \). The discriminant can tell you:

• If it's positive, the quadratic has two distinct real roots.
• If it's zero, the quadratic has exactly one real root, a repeated root.
• If it's negative, the quadratic has no real roots but two complex roots.

For the quadratic \( x^2 + 2x - 1 \) in our task, the discriminant computes to \( 8 \) (since \( 4 + 4 = 8 \)), indicating two distinct real roots. Knowing this before applying the quadratic formula provides a useful check on what kind of solutions to expect.