Problem 25 - Algebra 2

Question

Find all of the zeros of each function. \(f(x)=x^{3}-5 x^{2}-7 x+51\)

Step-by-Step Solution

Answer

The zeros are \( x = -3 \), \( 4 + i \), and \( 4 - i \).

Step 1: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial \( f(x) = x^3 - 5x^2 - 7x + 51 \) is a factor of the constant term divided by a factor of the leading coefficient. The constant term is 51, and the leading coefficient is 1. Therefore, the possible rational roots are the factors of 51: \( \pm 1, \pm 3, \pm 17, \pm 51 \).

Step 2: Test Potential Rational Roots Using Synthetic Division

Begin testing the possible rational roots to find any actual roots using synthetic division. Start with possible roots like \( x = 1, -1, 3, -3, 17, \) and \( -17 \). When tested with \( x = 3 \), perform the synthetic division and see if the remainder is zero.Setting up synthetic division for \( x = 3 \):\[\begin{array}{r|rrrr}3 & 1 & -5 & -7 & 51 \ & & 3 & -6 & -39 \\hline & 1 & -2 & -13 & 12 \\end{array}\]The remainder is not zero; thus, \( x = 3 \) is not a root. Continue testing potential roots.

Step 3: Identify a Rational Root

Continue testing potential rational roots until the remainder is zero. When \( x = -3 \) is tested, the remainder is zero using synthetic division:\[\begin{array}{r|rrrr}-3 & 1 & -5 & -7 & 51 \ & & -3 & 24 & -51 \\hline & 1 & -8 & 17 & 0 \\end{array}\]The remainder is zero; therefore, \( x = -3 \) is a root. This division leaves us with the quadratic \( x^2 - 8x + 17 \).

Step 4: Solve the Quadratic for Remaining Roots

Use the quadratic formula to find the roots of \( x^2 - 8x + 17 \). The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For \( x^2 - 8x + 17 \), \( a = 1, b = -8, c = 17 \). Plug these values into the formula:\[x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \times 1 \times 17}}{2 \times 1} \]\[x = \frac{8 \pm \sqrt{64 - 68}}{2} \]\[x = \frac{8 \pm \sqrt{-4}}{2} \]\[x = 4 \pm i \]The remaining solutions are complex: \( x = 4 + i \) and \( x = 4 - i \).

Step 5: Conclusion

The zeros of the polynomial function \( f(x) = x^3 - 5x^2 - 7x + 51 \) are \( x = -3 \), \( 4 + i \), and \( 4 - i \).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool for finding potential rational roots of polynomial equations. It states that any rational solution of a polynomial equation with integer coefficients will be a fraction, \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

When faced with a polynomial like \( f(x) = x^3 - 5x^2 - 7x + 51 \), you start by identifying its constant term, which is 51. The leading coefficient is 1.

• The factors of 51 are: \( \pm 1, \pm 3, \pm 17, \pm 51 \).
• The factors of 1 are just \( \pm 1 \).

Therefore, the possible rational roots we test are: \( \pm 1, \pm 3, \pm 17, \pm 51 \). Knowing this theorem allows you to narrow down your options and find roots more efficiently by only considering these specific values.

Synthetic division

Synthetic division offers a simplified way to divide polynomials, especially when dealing with possible roots. It helps us quickly determine if a test value is indeed a root of the polynomial.

To perform synthetic division, you will:

• Write the coefficients of the polynomial in a row.
• Place the test root on the left.
• Follow a pattern of multiplication and addition.

For example, testing \( x = -3 \) with \( f(x) = x^3 - 5x^2 - 7x + 51 \), you arrange it like this:
\[\begin{array}{r|rrrr}-3 & 1 & -5 & -7 & 51 \ & & -3 & 24 & -51 \\hline & 1 & -8 & 17 & 0 \\end{array}\]
The zero remainder signifies \( -3 \) is indeed a root. The result also gives you a simpler equation to solve: \( x^2 - 8x + 17 \). Synthetic division is much quicker and less prone to mistakes than traditional long division.

Quadratic formula

The quadratic formula is crucial for finding the roots of quadratic equations of the form \( ax^2 + bx + c = 0 \). It is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula allows you to determine the roots quickly, whether they are real or complex.

In our exercise, after using synthetic division, we arrived at the quadratic \( x^2 - 8x + 17 \). Here,

• \( a = 1 \)
• \( b = -8 \)
• \( c = 17 \)

Substituting these into the quadratic formula, we solve:
\[x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \times 1 \times 17}}{2 \times 1} \]
\[x = \frac{8 \pm \sqrt{-4}}{2} \]
This calculation reveals complex solutions: \( x = 4 + i \) and \( x = 4 - i \). Whenever the expression under the square root (discriminant) is negative, the roots will be complex, involving imaginary numbers.

Complex numbers

Complex numbers are essential when solving polynomial equations that do not have real roots. A complex number has a real part and an imaginary part, noted as \( a + bi \), where \( i \) is the imaginary unit with the property that \( i^2 = -1 \).

In the quadratic formula, when the discriminant (the part under the square root) is negative, the solutions are complex. This situation is precisely what we encountered with the quadratic \( x^2 - 8x + 17 \).

• The discriminant was \(-4\), leading to complex roots.
• Our solutions were \( 4 + i \) and \( 4 - i \), showcasing the imaginary unit \( i \).

Understanding complex numbers expands the ability to find all types of roots in polynomials, not just the real ones. They are vital in various mathematical and engineering fields, providing solutions beyond what real numbers can offer.