Problem 55 - Algebra and Trigonometry

Question

Show that the given value(s) of \(c\) are zeros of \(P(x)\), and find all other zeros of \(P(x)\). \(P(x)=x^{3}-x^{2}-11 x+15, \quad c=3\)

Step-by-Step Solution

Answer

The zeros of \( P(x) \) are \( x = 3, -1 + \sqrt{6}, -1 - \sqrt{6} \).

Step 1: Verify Given Zero

To verify if \( c = 3 \) is a zero of \( P(x) = x^3 - x^2 - 11x + 15 \), substitute \( x = 3 \) into \( P(x) \). Calculate: \( P(3) = 3^3 - 3^2 - 11 \times 3 + 15 \). This simplifies to \( 27 - 9 - 33 + 15 = 0 \). Since \( P(3) = 0 \), \( c = 3 \) is indeed a zero of \( P(x) \).

Step 2: Perform Polynomial Division

Next, use polynomial long division to factor \( P(x) \) by \( x - 3 \), since \( x = 3 \) is already a zero. Divide \( x^3 - x^2 - 11x + 15 \) by \( x - 3 \). The quotient will give us a polynomial of degree 2. Performing the division, we find it is \( x^2 + 2x - 5 \).

Step 3: Factor the Quadratic

To find other zeros of \( P(x) \), we need to solve \( x^2 + 2x - 5 = 0 \). Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = -5 \). Calculate the discriminant: \( b^2 - 4ac = 2^2 - 4 \times 1 \times (-5) = 4 + 20 = 24 \). So, \( x = \frac{-2 \pm \sqrt{24}}{2} \).

Step 4: Simplify the Roots

Simplify \( \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6} \). Thus, the solutions become \( x = \frac{-2 \pm 2\sqrt{6}}{2} \). Simplify further to get \( x = -1 \pm \sqrt{6} \). These are the two additional zeros of \( P(x) \).

Step 5: Conclusion

The zeros of the polynomial \( P(x) = x^3 - x^2 - 11x + 15 \) are \( x = 3, x = -1 + \sqrt{6}, \) and \( x = -1 - \sqrt{6} \).

Key Concepts

Zero of a Polynomial

The zero of a polynomial is a special value of x that makes the whole polynomial equal zero. In other words, if you substitute this value into the polynomial equation, the outcome will be zero.
When finding zeros, we often say we are solving the polynomial equation, or finding where the polynomial crosses the x-axis. For example, if we take the polynomial \( P(x) = x^3 - x^2 - 11x + 15 \) and substitute \( x = 3 \), calculating \( P(3) \), we find that it equals zero, proving that \( x = 3 \) is indeed a zero of this polynomial.
Remember, zeros can also be called roots of the polynomial, and they are crucial for graphing and understanding the behavior of polynomial functions.

Polynomial Division

Polynomial division is a process used to divide a polynomial by another polynomial of lower degree, similar to long division with numbers. When we know a certain value is a zero, we can use this method to simplify the polynomial.
To understand this, imagine you have a third-degree polynomial like \( P(x) = x^3 - x^2 - 11x + 15 \). Given that \( x = 3 \) is a zero, we can factor \( P(x) \) by dividing it by \( x - 3 \).
Through polynomial long division, dividing this specific polynomial by \( x - 3 \), we determine the quotient, which in this case is \( x^2 + 2x - 5 \). This quotient is key as it reveals the remaining factors of the original polynomial after removing the factor for the known zero.

Quadratic Formula

The quadratic formula is a universal method for finding zeros of quadratic equations, which are polynomials of degree two. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). By plugging the coefficients \( a \), \( b \), and \( c \) into this formula, you can find the roots.The quadratic formula states:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

For a quadratic derived from a divided polynomial, say \( x^2 + 2x - 5 \), you substitute \( a = 1 \), \( b = 2 \), \( c = -5 \) into the formula. This will provide the solutions that represent the remaining zeros of the original polynomial, which in this example are \( x = -1 \pm \sqrt{6} \).
The quadratic formula is not only efficient but also becomes indispensable when factoring is impractical or impossible.

Discriminant Calculation

Discriminant calculation involves determining the discriminant from a quadratic equation, which helps us analyze the nature of its roots without solving it completely. The discriminant is the part under the square root in the quadratic formula, \( b^2 - 4ac \).
This value reveals the nature of the roots:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is one real root with multiplicity two (repeated root).
• If \( b^2 - 4ac < 0 \), there are no real roots (the roots are complex).

For \( x^2 + 2x - 5 \), calculate the discriminant as \( 2^2 - 4 \times 1 \times (-5) = 4 + 20 = 24 \), indicating two distinct real roots exist.
Understanding the discriminant can offer insight into the graph's behavior, as well as provide preliminary information about the types of solutions for the polynomial.