Problem 56 - Algebra and Trigonometry

Question

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=-x^{4}+10 x^{2}+8 x-8 $$

Step-by-Step Solution

Answer

The real zeros are found using substitution and factoring; graph behavior is studied next.

Step 1: Identify the Polynomial Type

The polynomial \(P(x) = -x^4 + 10x^2 + 8x - 8\) is a quartic polynomial because the highest power of \(x\) is 4. It might have up to four real roots.

Step 2: Substitute for Possible Rational Zeros

Use the Rational Root Theorem to test possible rational zeros. The possible rational zeros are the factors of the constant term \(-8\) divided by the factors of the leading coefficient \(-1\). This gives the candidates \(\pm 1, \pm 2, \pm 4, \pm 8\).

Step 3: Test Each Possible Zero

Substitute each possible zero into \(P(x)\) to check if it equals zero. Evaluate \(P(1), P(-1), P(2), P(-2), P(4), P(-4), P(8), P(-8)\) to find which, if any, are roots.

Step 4: Factor the Polynomial (If Possible)

If you find any zeros from substitution, use those to factor the polynomial further. Typically, you'll employ polynomial division to factor the original polynomial using the roots found.

Step 5: Solve for Other Zeros

If factoring is difficult, or if no rational zeros are found, consider trying to factor using techniques like completing the square or synthetic division to simplify and find remaining roots.

Step 6: Graph the Polynomial

Use the roots found and the leading coefficient to sketch the graph of \(P(x)\). The graph of a degree 4 polynomial will have a maximum of three turning points and will approach negative infinity as \(x\) approaches positive or negative infinity because of the negative leading coefficient.

Step 7: Analyze the Graph

Use the zeros found and the shape of the graph to verify the nature of the function. Confirm that the behavior at each zero is consistent with the multiplicity of the root found and check if the graph agrees with the end behavior described in Step 6.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a valuable tool for finding possible rational zeros of a polynomial. It suggests that any rational solution of a polynomial equation with integer coefficients, say \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0\), is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\).
For the polynomial \(P(x) = -x^4 + 10x^2 + 8x - 8\), the possible rational roots are derived from the factors of the constant term \(-8\) and the leading coefficient \(-1\).

• Factors of \(-8\) are \(\pm 1, \pm 2, \pm 4, \pm 8\).
• The leading coefficient \(-1\) has factors \(\pm 1\).

Thus, the possible rational zeros are \(\pm 1, \pm 2, \pm 4, \pm 8\). By testing these candidates, we determine whether they are actual roots of the polynomial.

Polynomial Division

Once a rational root is found, it may be necessary to perform polynomial division to factor the polynomial further. Polynomial division allows us to divide one polynomial by another, simplifying it, and is akin to long division with numbers.
Given a polynomial \(P(x)\) and a discovered root \(r\), it can be expressed as \(P(x) = (x - r)Q(x) + R\), where \(Q(x)\) is the quotient and \(R\) is the remainder. If \(r\) is a true root, the remainder \(R\) should be zero, indicating that \((x - r)\) is a factor of \(P(x)\).
Successfully dividing \(P(x)\) by \((x - r)\) helps break down the polynomial into simpler components, each of which can be analyzed for further roots. This step is crucial for systematically finding all zeros of the polynomial.

Real Zeros

Real zeros of a polynomial are the values of \(x\) for which the polynomial equals zero. These zeros correspond to the points where the graph of the polynomial crosses the x-axis.
In the process of solving\(P(x) = -x^4 + 10x^2 + 8x - 8\), we determined the possible rational zeros and used polynomial division to simplify the polynomial. The real zeros can then be identified clearly, contributing to understanding the polynomial's overall structure.
The number of real zeros a polynomial can have is at most equal to its degree. In this case, a quartic polynomial (degree 4) can have up to four real zeros. Each zero indicates an intersection point or touching point on the x-axis in the graph.

Graphing Polynomials

Graphing a polynomial gives a visual representation of its behavior, uncovering zeros, turning points, and end behavior. For the polynomial \(P(x) = -x^4 + 10x^2 + 8x - 8\), the graph has characteristic features of a quartic polynomial.
The end behavior is determined by the leading term \(-x^4\), implying that as \(x\) approaches infinity or negative infinity, \(P(x)\) approaches negative infinity. This means the tails of the graph will point downwards on both ends.
Real zeros, found using steps like testing rational roots and performing polynomial division, further refine the graph. With up to three turning points, the graph’s shape portrays maximum and minimum heights, which influence how the curve flows between zeros.
Ultimately, by understanding the graph's structure, its interaction with the x-axis, and determining where it turns, a comprehensive picture of polynomial behavior emerges.