Problem 16
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. $$ f(x)-x^{3}-4 x^{2}+8 x-5 $$
Step-by-Step Solution
Verified Answer
The zeros of the polynomial \(f(x) = x^{3} - 4x^{2} + 8x - 5\) are \(1, (5 + sqrt{5})/2, (5 - sqrt{5})/2\).
1Step 1: List all possible rational zeros
According to the Rational Root Theorem, if a polynomial has a rational root \(p/q\), then \(p\) is a factor of the constant term (in this case -5) and \(q\) is a factor of the coefficient for the highest degree (in this case 1). Therefore we have the possible rational roots: \(\pm1, \pm5\).
2Step 2: Use synthetic division to test possible rational zeros
Next, perform synthetic division on the original polynomial using each of the possible rational zeros until finding an actual zero. When using \(-1\) as divider, we obtain a remainder of -8. When using \(1\) as divider, we obtain a remainder of 0, this confirms \(1\) is a root of the polynomial.
3Step 3: Use quotient from Step 2 to find the remaining zeros
The quotient from the division in stage 2 is \(x^2 - 5x + 5\), the remaining zeros of the polynomial can thus be obtained by setting this equation equal to zero and solving for \(x\). To do so, apply the quadratic formula, \(x = [-(-5) \pm sqrt{(-5)^2 - 4*1*5}]/2*1 = (5 \pm sqrt{5})/2, therefore the two other roots are \((5 \pm sqrt{5})/2\).
Key Concepts
Synthetic DivisionPolynomial RootsQuadratic Formula
Synthetic Division
Synthetic division is a method used to divide a polynomial by a binomial of the form \(x - c\) where \(c\) is a constant. This technique is especially useful when dealing with higher degree polynomials and is commonly applied to find zeros or to simplify polynomials.
To perform synthetic division, one should first write down the coefficients of the polynomial in descending order of the powers of \(x\). After this, draw a horizontal line and write the value of \(c\) to the left. The process involves bringing the leading coefficient down, multiplying it by \(c\), placing the result underneath the next coefficient, and then adding the two numbers in that column. This sequence of actions is repeated until reaching the last coefficient. The numbers that appear on the bottom row, aside from the last one, are the coefficients of the quotient polynomial, and the last number is the remainder. If the remainder is zero, then \(x - c\) is a factor of the polynomial.
Using synthetic division helps in testing possible rational zeros quickly, as shown in the textbook exercise. If the remainder is not zero, the division indicates that the tested number is not a root of the polynomial. Testing each possible root as described helps converge on a correct factor of the polynomial efficiently.
To perform synthetic division, one should first write down the coefficients of the polynomial in descending order of the powers of \(x\). After this, draw a horizontal line and write the value of \(c\) to the left. The process involves bringing the leading coefficient down, multiplying it by \(c\), placing the result underneath the next coefficient, and then adding the two numbers in that column. This sequence of actions is repeated until reaching the last coefficient. The numbers that appear on the bottom row, aside from the last one, are the coefficients of the quotient polynomial, and the last number is the remainder. If the remainder is zero, then \(x - c\) is a factor of the polynomial.
Using synthetic division helps in testing possible rational zeros quickly, as shown in the textbook exercise. If the remainder is not zero, the division indicates that the tested number is not a root of the polynomial. Testing each possible root as described helps converge on a correct factor of the polynomial efficiently.
Polynomial Roots
Polynomial roots are the values of \(x\) that make the polynomial function equal to zero. These are also referred to as 'zeros' or 'solutions' to the polynomial equation. Finding the roots of a polynomial is central to many areas of mathematics and applied science because it allows for understanding the behavior of the function.
The Rational Zeros Theorem provides a systematic way to list all possible rational zeros of a polynomial function. These potential zeros are based on the factors of the constant term and the factors of the leading coefficient. However, not all listed values will be actual zeros; they must be tested. This can be done efficiently by applying synthetic division for each possible zero. Once an actual zero is found, it can often be used to factor the polynomial further or to reduce the degree of the polynomial, making it easier to find additional roots.
In the exercise provided, the Rational Zeros Theorem is used initially to list the potential zeros of the polynomial. Following this, synthetic division was performed to test these candidates and identify the actual zeros, illustrating the practical application of these concepts in problem-solving.
The Rational Zeros Theorem provides a systematic way to list all possible rational zeros of a polynomial function. These potential zeros are based on the factors of the constant term and the factors of the leading coefficient. However, not all listed values will be actual zeros; they must be tested. This can be done efficiently by applying synthetic division for each possible zero. Once an actual zero is found, it can often be used to factor the polynomial further or to reduce the degree of the polynomial, making it easier to find additional roots.
In the exercise provided, the Rational Zeros Theorem is used initially to list the potential zeros of the polynomial. Following this, synthetic division was performed to test these candidates and identify the actual zeros, illustrating the practical application of these concepts in problem-solving.
Quadratic Formula
The quadratic formula is used to find the roots of a quadratic equation, which is a second-degree polynomial equation in the form \(ax^{2}+bx+c=0\). The formula states that the roots of a quadratic equation can be found using the following expression: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\). The symbol \(\pm\) indicates that there are usually two solutions: one where you add the square root term, and one where you subtract it.
This formula is crucial because it provides an exact method to find roots of any quadratic equation, regardless of whether the roots are real or complex numbers. By applying the quadratic formula to the quotient obtained from synthetic division with a zero remainder, as illustrated in the exercise, the remaining roots of the original polynomial are revealed.
The exercise solution demonstrates using the quadratic formula after reducing the original third-degree polynomial to a quadratic one, via synthetic division by one of its roots. The formula's application yields the final two roots and thus completely solves the polynomial equation, showcasing a typical sequence used to tackle higher-degree polynomial problems.
This formula is crucial because it provides an exact method to find roots of any quadratic equation, regardless of whether the roots are real or complex numbers. By applying the quadratic formula to the quotient obtained from synthetic division with a zero remainder, as illustrated in the exercise, the remaining roots of the original polynomial are revealed.
The exercise solution demonstrates using the quadratic formula after reducing the original third-degree polynomial to a quadratic one, via synthetic division by one of its roots. The formula's application yields the final two roots and thus completely solves the polynomial equation, showcasing a typical sequence used to tackle higher-degree polynomial problems.
Other exercises in this chapter
Problem 16
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Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$ \frac{2 x^{5}-8 x^{4}+2 x^{3}+x^{2}}{2 x^{3}+1} $$
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Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$f(x)--2 x^{2}+8 x-1$$
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