Problem 55 - Algebra and Trigonometry

Question

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=x^{4}-5 x^{3}+6 x^{2}+4 x-8 $$

Step-by-Step Solution

Answer

The real zero of \(P\) is \(x=2\). The graph rises on both ends and crosses the x-axis at \(x=2\).

Step 1: Verify possible rational roots

To find the real zeros of the polynomial \(P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8\), use the Rational Root Theorem. It states that any rational root, \(p/q\), is such that \(p\) divides the constant term, \(-8\), and \(q\) divides the leading coefficient, \(1\). The possible rational roots are: ±1, ±2, ±4, ±8. Test these by substituting into \(P(x)\).

Step 2: Test rational roots

Substitute each candidate from Step 1 into \(P(x)\):- \(P(1) = 1^4 - 5(1)^3 + 6(1)^2 + 4(1) - 8 = -2\) - \(P(-1) = (-1)^4 - 5(-1)^3 + 6(-1)^2 + 4(-1) - 8 = -6\)- \(P(2) = 2^4 - 5(2)^3 + 6(2)^2 + 4(2) - 8 = 0\) **Root found: \(x=2\)**- \(P(-2)\), \(P(4)\), \(P(-4)\), \(P(8)\), \(P(-8)\) do not satisfy \(P(x)=0\).Thus, \(x = 2\) is a root.

Step 3: Perform synthetic division

To factor \(P(x)\) by \((x - 2)\), perform synthetic division:1. Write coefficients: 1, -5, 6, 4, -8.2. Bring down 1.3. Multiply by 2, place result under next coefficient.4. Add.5. Repeat.Remainder is 0; new coefficients are \(1, -3, 0, 4\), so \(P(x)=(x - 2)(x^3 - 3x^2 + 4)\).

Step 4: Solve reduced polynomial

Factor \(x^3 - 3x^2 + 4\). Guess and test other rational roots or use derivatives to find critical points. Using the quadratic formula for \(x^2 - 3x + 4\) results in non-real roots, but further testing shows that there are no other rational roots.The remaining cubic polynomial cannot be reduced using simple factorization or synthetic division.

Step 5: Graph the polynomial

Sketch the polynomial graph using found roots and behavior at infinity. The polynomial has a root at \(x=2\). The end behavior is determined by \(x^4\), so it rises on both ends. There are no other real roots from \(x^3 - 3x^2 + 4\), so the graph doesn't cross the x-axis elsewhere.

Key Concepts

Synthetic Division

When you want to divide a polynomial by a binomial like \((x - 2)\), synthetic division is your go-to method. It's a simplified form of polynomial division that is easier than long division. Here's how synthetic division works in simple terms:

• First, write down the coefficients of the polynomial. For \(x^4 - 5x^3 + 6x^2 + 4x - 8\), the coefficients are 1, -5, 6, 4, and -8.
• Next, write the root used for division on the side. If dividing by \((x - 2)\), write 2.
• Bring down the first coefficient.
• Multiply the root by the brought-down number, and place the result under the next coefficient.
• Add down the column.
• Repeat multiply and add steps until all coefficients are processed.

The remainder from synthetic division should be zero if \(x - 2\) is a true factor. The new coefficients give you a transformed polynomial, showing how \(P(x)\) breaks down into simpler polynomials like \((x - 2)(x^3 - 3x^2 + 4)\).
This method is quick and helps confirm if a potential root divides the polynomial evenly.

Rational Root Theorem

The Rational Root Theorem is an essential tool when finding the roots of polynomials. Here's the core idea: to find any rational zeros!

• Look at the constant term in the polynomial. For \(x^4 - 5x^3 + 6x^2 + 4x - 8\), the constant is -8.
• Identify the leading coefficient, which is 1 in this polynomial.
• The theorem tells us that any rational zero can be written as \(\frac{p}{q}\), where \(p\) is a factor of the constant term (-8) and \(q\) is a factor of the leading coefficient (1).
• This means possible rational roots are ±1, ±2, ±4, and ±8.

By testing these values in the polynomial, any that result in zero are actual roots. The process narrows down potential zeros before diving into practical testing or factoring, saving time and effort. When you know what might work, it's easier to unravel the mysteries of the polynomial.

Polynomial Factorization

Polynomial factorization involves breaking down a complex polynomial like \(x^4 - 5x^3 + 6x^2 + 4x - 8\) into simpler factors. Here's how it can simplify finding zeros:

• Start by finding real roots using tools like the Rational Root Theorem. For our polynomial, we identified that \(x = 2\) is a root.
• Use synthetic division to divide out the factor \((x - 2)\). This simplifies the polynomial to a lower degree.
• You are left with \(x^3 - 3x^2 + 4\), a cubic polynomial. The goal is to factor it further if possible.

While our example didn't yield more real factors through basic methods, understanding polynomial factorization is key to solving. It helps identify all the real zeros and provides an efficient way to deal with higher degree polynomials by breaking them down step-by-step.
When successful, polynomial factorization turns complex equations into manageable pieces, revealing all real zeros systematically.