Problem 14
Question
In Exercises \(9-16\) a. List all possible rational zeros. b. Use synthetic division to test the possible rational zeros and find an actual zero. c. Use the quotient from part (b) to find the remaining zeros of the polynomial function. $$ f(x)-2 x^{3}+x^{2}-3 x+1 $$
Step-by-Step Solution
Verified Answer
The zeros of the polynomial function \(f(x) = -2x^{3}+x^{2}-3x+1\) are \(x=-1/2, -1+\sqrt{3}, -1-\sqrt{3}\).
1Step 1: Listing all possible rational zeros
Utilize the Rational Root Theorem which states that if a polynomial has a rational zero \(p/q\), then \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. For the given \(f(x) = -2x^{3}+x^{2}-3x+1\), the possible rational zeros will be \(\pm\) factors of \(1\) over \(\pm\) factors of \(2\), which gives us \(\pm1, \pm1/2\).
2Step 2: Testing the possible rational zeros
The possible zeros \(\pm1, \pm1/2\) will be tested using synthetic division. Upon doing the synthetic division, it will turn out that only \(x=-1/2\) is a zero, as it yields a remainder of zero.
3Step 3: Finding the remaining zeros
After confirming \(x=-1/2\) as a zero, the quotient from the synthetic division \(-4x^{2}-2x+2\) can be equated to zero and solved (using either factoring, completing the square, or quadratic formula) for the remaining zeros of the function. Solving this results in two additional zeroes, which are \(x=-1 \pm \sqrt{3}\).
Key Concepts
Rational Root TheoremSynthetic DivisionZeroes of Polynomial FunctionsQuadratic Formula
Rational Root Theorem
The Rational Root Theorem is a handy guideline in algebra. It helps to identify potential rational zeroes of polynomial functions. This theorem states that if a polynomial has a rational root \( \frac{p}{q} \), then \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
For example, consider the polynomial \( f(x) = -2x^3 + x^2 - 3x + 1 \). Here, the constant term is \( 1 \), and the leading coefficient is \( -2 \). Using the Rational Root Theorem, we consider all the factors of \( 1 \) (which are \( \pm1 \)) and the factors of \( -2 \) (which are \( \pm1, \pm2 \)).
For example, consider the polynomial \( f(x) = -2x^3 + x^2 - 3x + 1 \). Here, the constant term is \( 1 \), and the leading coefficient is \( -2 \). Using the Rational Root Theorem, we consider all the factors of \( 1 \) (which are \( \pm1 \)) and the factors of \( -2 \) (which are \( \pm1, \pm2 \)).
- This leads to potential rational zeroes: \( \pm1 \) and \( \pm\frac{1}{2} \).
Synthetic Division
Synthetic division is a simplified method of polynomial division, specifically meant for dividing by linear factors. It is useful in testing potential rational zeroes derived from the Rational Root Theorem.
Unlike long division, synthetic division involves fewer steps and is less cumbersome. It's particularly efficient when you need to check multiple potential roots.
In the exercise relating to \( f(x) = -2x^3 + x^2 - 3x + 1 \), we utilize synthetic division to test the potential zeroes: \( \pm1 \) and \( \pm\frac{1}{2} \). Among these, \( x = -\frac{1}{2} \) comes out as an actual zero since the process yields a remainder of zero.
Once a real zero is found, synthetic division also provides the quotient polynomial, which helps in finding the remaining zeroes.
Unlike long division, synthetic division involves fewer steps and is less cumbersome. It's particularly efficient when you need to check multiple potential roots.
In the exercise relating to \( f(x) = -2x^3 + x^2 - 3x + 1 \), we utilize synthetic division to test the potential zeroes: \( \pm1 \) and \( \pm\frac{1}{2} \). Among these, \( x = -\frac{1}{2} \) comes out as an actual zero since the process yields a remainder of zero.
Once a real zero is found, synthetic division also provides the quotient polynomial, which helps in finding the remaining zeroes.
Zeroes of Polynomial Functions
Zeroes of polynomial functions are the values of \( x \) for which the polynomial evaluates to zero. These zeroes are critical as they give insight into the function's graph by indicating where it intersects the x-axis.
In the function \( f(x) = -2x^3 + x^2 - 3x + 1 \), zeroes can be found by applying methods like synthetic division and further factoring the resulting quotient.
In the function \( f(x) = -2x^3 + x^2 - 3x + 1 \), zeroes can be found by applying methods like synthetic division and further factoring the resulting quotient.
- Initially, using synthetic division, one of the zeroes \( x = -\frac{1}{2} \) is found.
- This discovery transforms the cubic polynomial into a quadratic through division.
- The remaining quadratic \( -4x^2 - 2x + 2 = 0 \) can then be tackled using other algebraic techniques.
Quadratic Formula
When a polynomial is reduced to a quadratic expression, the quadratic formula is a strong tool for finding zeroes. The formula, \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \), solves the equation \( ax^2 + bx + c = 0 \).
The quadratic formula is especially useful when other simple methods such as factoring are not feasible.
For the quadratic \( -4x^2 - 2x + 2 = 0 \), extracted from our primary polynomial using synthetic division, it's efficient to apply the quadratic formula.
The quadratic formula is especially useful when other simple methods such as factoring are not feasible.
For the quadratic \( -4x^2 - 2x + 2 = 0 \), extracted from our primary polynomial using synthetic division, it's efficient to apply the quadratic formula.
- Here, \( a = -4 \), \( b = -2 \), and \( c = 2 \).
- Applying these values to the formula provides the zeroes: \( x = -1 \pm \sqrt{3} \).
Other exercises in this chapter
Problem 14
Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ 6 x^
View solution Problem 14
Write an equation that expresses each relationship. Then solve the equation for y. \(x\) varies directly as the cube root of \(z\) and inversely as \(y\)
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Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$ \frac{x^{4}+2 x^{3}-4 x^{2}-5 x-6}{x^{2}+x-2} $$
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Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$f(x)-3 x^{2}-12 x+1$$
View solution