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Magazine Chapter 4: Problem 45
Factor the polynomial function \(f(x) .\) Then solve the equation \(f(x)=0\) $$f(x)=x^{4}-7 x^{3}+9 x^{2}+27 x-54$$
Short Answer
Expert verified
The roots are \(x = 3\) (with multiplicity 3) and \(x = -2\).
Step by step solution
01
- Look for rational roots
To find potential roots, apply the Rational Root Theorem. The possible rational roots are the factors of the constant term (-54) divided by the factors of the leading coefficient (1). This gives us potential roots such as 卤1, 卤2, 卤3, 卤6, 卤9, 卤18, 卤27, 卤54.
02
- Test potential roots
Test the potential rational roots by substituting them into the polynomial. For example, substituting 2: \[ f(2) = 2^4 - 7(2)^3 + 9(2)^2 + 27(2) - 54 = 16 - 56 + 36 + 54 - 54 = -4 \]. Try different values until a root is found. By testing, it can be discovered that 3 is a root.
03
- Factor out the root
Once a root is found (e.g., x = 3), use synthetic division to factor out \((x - 3)\) from the polynomial \(f(x)\). Performing synthetic division of \(x^4 - 7x^3 + 9x^2 + 27x - 54\) by \(x - 3\) will yield \((x - 3)(x^3 - 4x^2 - 3x + 18)\).
04
- Further factor the quotient
Now, factor the cubic polynomial \(x^3 - 4x^2 - 3x + 18\). Continue testing for additional roots. Suppose we find another root like \(x = 3\), we can use synthetic division again: Divide \(x^3 - 4x^2 - 3x + 18\) by \(x - 3\) to get \((x - 3)(x^2 - x - 6)\).
05
- Factor the quadratic polynomial
Next, factor the quadratic polynomial \(x^2 - x - 6\). This can be factored into \((x - 3)(x + 2)\).
06
- Combine all factors
Combine all the factors found. The original polynomial \(f(x)\) can be expressed as \[(x - 3)^3 (x + 2)\].
07
- Solve the equation \(f(x) = 0\)
To solve \(f(x) = 0\), set each factor equal to zero:\((x - 3)^3 = 0\) and \(x + 2 = 0\). Solving these gives the roots:\[x = 3\] (with multiplicity 3) and \[x = -2\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rational Root Theorem
The Rational Root Theorem is a tool that helps identify possible rational roots of a polynomial equation. It states that any rational root of a polynomial, written as a ratio \(\frac{p}{q}\), will have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient. For the polynomial \(f(x) = x^4 - 7x^3 + 9x^2 + 27x - 54\), the constant term is \(-54\) and the leading coefficient is \(1\). Thus, the possible rational roots are the factors of \(-54\), which are \(\begin{pmatrix} \pm1, \pm2, \pm3, \pm6, \pm9, \pm18, \pm27, \pm54 \end{pmatrix}\). Testing these potential roots helps us narrow down the actual roots of the polynomial.
Synthetic Division
Synthetic division is a simplified form of polynomial division, used primarily to divide polynomials by binomials of the form \(x - c\). It helps in factoring polynomials after we identify a root. To use synthetic division, write the coefficients of the polynomial in a row. Then, use the identified root to perform the division, which simplifies the polynomial step by step. For example, using the identified root \(x = 3\) on the polynomial \(x^4 - 7x^3 + 9x^2 + 27x - 54\) using synthetic division yields the quotient \(x^3 - 4x^2 - 3x + 18\). Continue this process as needed to factor the polynomial further.
Cubic Polynomial
After finding a root and performing synthetic division, we might end up with a cubic polynomial. For our polynomial \((x - 3)(x^3 - 4x^2 - 3x + 18)\), the next step is to factor this cubic polynomial. Using the Rational Root Theorem again and synthetic division, we locate additional roots. After we find another root, say \(x = 3\) again, perform synthetic division on the cubic polynomial to get a quadratic polynomial. The goal is to reduce the polynomial to a degree where factoring becomes simpler.
Quadratic Polynomial
Eventually, synthetic division reduces higher-degree polynomials to quadratic polynomials, which are easier to factor. The quadratic polynomial \(x^2 - x - 6\) can be factored manually. We look for factors of the constant term, \(-6\), that add up to the coefficient of the linear term, \(-1\). Here, \(-3\) and \(+2\) fit the bill. Thus, the quadratic polynomial factors into \( (x - 3)(x + 2) \). These factors help further break down the original polynomial into simpler, solvable parts.
Multiplicity of Roots
Multiplicity of roots refers to the number of times a particular root appears in the factored form of a polynomial. In our case, the original polynomial \(f(x) = x^4 - 7x^3 + 9x^2 + 27x - 54\) factors into \((x - 3)^3(x + 2)\). This shows that \(x = 3\) is a root with a multiplicity of 3, meaning it appears three times. On the other hand, \(x = -2\) is a root with a multiplicity of 1. To solve \(f(x) = 0\), set each factor equal to zero. Hence, the solutions are \(x = 3\) and \( x = -2\).
