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Magazine Chapter 4: Problem 41
Find the zeros of the polynomial function and state the multiplicity of each. $$f(x)=2 x^{3}-x^{2}-8 x+4$$
Short Answer
Expert verified
The zeros are \( x = 2 \), \( x = \frac{1}{2} \), and \( x = -2 \), each with multiplicity 1.
Step by step solution
01
- Find potential rational roots
Use the Rational Root Theorem to find potential rational roots. The Rational Root Theorem states that any rational root of the polynomial equation \[ a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 = 0 \] is a fraction \( \frac{p}{q} \) 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 \( f(x) = 2x^3 - x^2 - 8x + 4 \), the constant term is 4 and the leading coefficient is 2. The factors of 4 are \{1, 2, 4\} and the factors of 2 are \{1, 2\}. Thus, the potential rational roots are \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2} \).
02
- Test potential rational roots using synthetic division
Test each potential rational root using synthetic division to see if it is an actual root of the polynomial. Start with \( x = 1 \): 1 | 2 -1 -8 4 | 2 1 -7 ---------------- 2 1 -7 -3Since the remainder is not 0, \( x = 1 \) is not a root. Next, try \( x = -1 \):-1 | 2 -1 -8 4 | -2 3 5 ---------------- 2 -3 -5 9Since the remainder is not 0, \( x = -1 \) is not a root. Continue this process for the rest of the potential rational roots.
03
- Identifying actual roots
Continue testing with synthetic division until finding an actual root. When testing \( x = 2 \): 2 | 2 -1 -8 4 | 4 6 -4 ---------------- 2 3 -2 0The remainder is 0, so \( x = 2 \) is a root. The quotient is \( 2x^2 + 3x - 2 \).
04
- Factor remaining polynomial
Factor the quadratic polynomial \( 2x^2 + 3x - 2 \). Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = 3 \), and \( c = -2 \). Calculate the discriminant: \[ \Delta = b^2 - 4ac = 3^2 - 4(2)(-2) = 9 + 16 = 25 \]. Thus, \( x = \frac{-3 \pm \sqrt{25}}{4} \) which simplifies to \( x = \frac{-3 \pm 5}{4} \). Therefore, \( x = \frac{2}{4} = \frac{1}{2} \) and \( x = \frac{-8}{4} = -2 \).
05
- State the roots and their multiplicities
The roots are \( x = 2 \), \( x = \frac{1}{2} \), and \( x = -2 \), and each has a multiplicity of 1 since each factor appears only once in the polynomial.
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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 useful tool for identifying potential rational zeros of a polynomial. According to this theorem, if a polynomial has any rational zeros, they must take the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. This theorem helps narrow down our search to a specific set of candidates.
synthetic division
Synthetic division is a streamlined way of dividing a polynomial by a binomial of the form \( x - c \). Unlike traditional long division, synthetic division involves fewer steps and calculations. It focuses on the coefficients of the polynomial and uses simple arithmetic to determine whether a given number is a root of the polynomial.
Steps to perform synthetic division:
Steps to perform synthetic division:
- Write down the coefficients of the polynomial.
- Write the suspected root to the far left.
- Carry out the synthetic division by following the pattern of multiplying and adding.
factoring polynomials
Once a root is found using synthetic division, the polynomial can be factored into smaller polynomials. This involves breaking down the original polynomial into simpler products of lower-degree polynomials. For example, given the root \( x = 2 \), the polynomial \( 2x^3 - x^2 - 8x + 4 \) can be factored into \( (x - 2)(2x^2 + 3x - 2) \).
Factoring helps simplify the polynomial further and makes it easier to solve or find more roots. It is a crucial step before using other solving techniques like the quadratic formula.
Factoring helps simplify the polynomial further and makes it easier to solve or find more roots. It is a crucial step before using other solving techniques like the quadratic formula.
quadratic formula
The quadratic formula is a reliable method for finding the roots of any quadratic equation: \[ ax^2 + bx + c = 0 \]. The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
This formula accounts for all possible roots, including real and complex numbers. It's derived from completing the square on the general form of a quadratic equation. By substituting the coefficients \( a \), \( b \), and \( c \) into the formula, we can easily find the roots. For instance, using \( 2x^2 + 3x - 2 \):
This formula accounts for all possible roots, including real and complex numbers. It's derived from completing the square on the general form of a quadratic equation. By substituting the coefficients \( a \), \( b \), and \( c \) into the formula, we can easily find the roots. For instance, using \( 2x^2 + 3x - 2 \):
- Calculate the discriminant: \[ \Delta = b^2 - 4ac \].
- Substitute into the quadratic formula to find \( x \) values.
