Problem 10 - Precalculus with Calculus Previews

Question

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=x^{4}+2 x^{3}-2 x^{2}-6 x-3 $$

Step-by-Step Solution

Answer

The rational zeros are \(-1\) repeated.

Step 1: Identify Potential Rational Zeros

According to the Rational Root Theorem, the potential rational zeros of a polynomial are given by the ratio of the factors of the constant term to the factors of the leading coefficient. For \( f(x) = x^4 + 2x^3 - 2x^2 - 6x - 3 \), the constant term is \(-3\) and the leading coefficient is \(1\). The factors of \(-3\) are \(\pm 1, \pm 3\) and the factors of \(1\) are \(\pm 1\). Therefore, the potential rational zeros are \(\pm 1, \pm 3\).

Step 2: Evaluate Potential Zeros in the Polynomial

Substitute each potential zero into the polynomial function to verify if it results in zero:- For \( f(1): 1^4 + 2 \cdot 1^3 - 2 \cdot 1^2 - 6 \cdot 1 - 3 = 1 + 2 - 2 - 6 - 3 = -8 eq 0 \)- For \( f(-1): (-1)^4 + 2(-1)^3 - 2(-1)^2 - 6(-1) - 3 = 1 - 2 - 2 + 6 - 3 = 0 \)- For \( f(3): 3^4 + 2 \cdot 3^3 - 2 \cdot 3^2 - 6 \cdot 3 - 3 = 81 + 54 - 18 - 18 - 3 = 96 eq 0 \)- For \( f(-3): (-3)^4 + 2(-3)^3 - 2(-3)^2 - 6(-3) - 3 = 81 - 54 - 18 + 18 - 3 = 24 eq 0 \)Hence, \(-1\) is a zero of the polynomial, and others are not.

Step 3: Perform Polynomial Division

Divide \(f(x)\) by \(x + 1\) since \(-1\) is a zero, using synthetic division or long division:- Synthetic division setup with \(-1\): Coefficients are \([1, 2, -2, -6, -3]\).- After division, the quotient is \(x^3 + x^2 - 3x - 3\) and the remainder is 0, indicating the division is exact.

Step 4: Find Zeros of the Quotient Polynomial

Now, solve the cubic polynomial \(x^3 + x^2 - 3x - 3\) to find more rational zeros:- Use the Rational Root Theorem again with potential zeros \(\pm 1, \pm 3\).- Evaluate: - \(f(1) = 1 + 1 - 3 - 3 = -4 eq 0\) - \(f(-1) = (-1)^3 + (-1)^2 - 3(-1) - 3 = -1 + 1 + 3 - 3 = 0\), thus \(-1\) is another root of the quotient polynomial.Divide \(x^3 + x^2 - 3x - 3\) by \(x + 1\), which gives \(x^2 - 3\) after synthetic division.

Step 5: Solve the Quadratic Polynomial

Now solve \(x^2 - 3 = 0\) using the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\).- Here, \(a = 1, b = 0, c = -3\).- The discriminant \(b^2 - 4ac = 0 - 4 \cdot 1 \cdot (-3) = 12\).- Thus, \(x = \frac{{0 \pm \sqrt{12}}}{2} = \pm \sqrt{3}\).- Since \(\sqrt{3}\) is irrational, there are no additional rational zeros from this step.

Key Concepts

Polynomial Division

Polynomial division is a technique used to divide polynomials, much like how you might divide numbers. It involves finding how many times a polynomial, known as the divisor, divides into another polynomial, known as the dividend, resulting in a quotient and sometimes a remainder. In the context of the exercise, once we found that \(-1\) is a zero of the polynomial using the Rational Root Theorem, we performed a division to simplify the polynomial. We divided \(f(x) = x^4 + 2x^3 - 2x^2 - 6x - 3\) by \(x + 1\), which was associated with the zero \(-1\). This division can be done using either long division or synthetic division, both of which would yield a quotient and possibly a remainder. Here, the division resulted in a cubic polynomial \(x^3 + x^2 - 3x - 3\) with no remainder, confirming \(x + 1\) is a factor of the original polynomial.

Rational Zeros

Rational zeros are zeroes of a polynomial that can be expressed as a simple fraction, or a rational number. When searching for rational zeros, the Rational Root Theorem is a valuable tool. This theorem states that any rational solution \(\frac{p}{q}\) of a polynomial equation with integer coefficients is a factor of the constant term \(p\), divided by a factor of the leading coefficient \(q\).For our given polynomial, \(f(x) = x^4 + 2x^3 - 2x^2 - 6x - 3\), the potential rational zeros can be \(\pm 1, \pm 3\). These are derived from the factors of \(-3\) (the constant term) and the factors of \(1\) (the leading coefficient).Substituting these potential zeros into the polynomial helps in verifying which ones actually zero the polynomial, like we found \(-1\) does.

Synthetic Division

Synthetic division is a shorthand method of dividing a polynomial by a linear binomial of the form \(x - c\). It's typically faster and simpler than long division. This process is particularly handy for testing potential zeros of a polynomial.In our exercise, once \(-1\) was identified as a rational zero, synthetic division was used. Using the coefficients \([1, 2, -2, -6, -3]\) corresponding to \(x^4 + 2x^3 - 2x^2 - 6x - 3\), synthetic division allowed us to simplify and divide the polynomial efficiently, confirming it with a quotient of \(x^3 + x^2 - 3x - 3\) and a remainder of zero, which indicates exact division.

Quadratic Polynomial

A quadratic polynomial is a polynomial equation of degree two, generally represented as \(ax^2 + bx + c\). It's typically solved using methods like factoring, completing the square, or the quadratic formula.In solving the original problem, after reducing the polynomial with synthetic division to find \(x^3 + x^2 - 3x - 3\), further division by \(x + 1\) resulted in a simpler quadratic polynomial: \(x^2 - 3\). Solving this involves setting it equal to zero: \(x^2 - 3 = 0\).Applying the quadratic formula, \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), yields solutions \(x = \pm \sqrt{3}\). Because \(\sqrt{3}\) is not a rational number, there are no additional rational zeros from this quadratic polynomial.