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Evaluate or simplify each expression without using a calculator. In e9x
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 95
Find all zeros of f(x) = x³ + 5x² – 8x + 2.
Guida verificata passo dopo passo1
Start by identifying the polynomial function: \(f(x) = x^{3} + 5x^{2} - 8x + 2\).
Use the Rational Root Theorem to list all possible rational zeros. These are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (2) and \(q\) divides the leading coefficient (1). So possible rational roots are \(\pm 1, \pm 2\).
Test each possible rational root by substituting into \(f(x)\) to see if it equals zero. For example, evaluate \(f(1)\), \(f(-1)\), \(f(2)\), and \(f(-2)\).
Once a root \(r\) is found, use polynomial division or synthetic division to divide \(f(x)\) by \((x - r)\), reducing the cubic to a quadratic.
Solve the resulting quadratic equation using factoring, completing the square, or the quadratic formula to find the remaining zeros.
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Polynomial Zeros
Zeros of a polynomial are the values of x for which the polynomial equals zero. Finding zeros involves solving the equation f(x) = 0, which helps identify the roots or x-intercepts of the polynomial function.
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Finding Zeros & Their Multiplicity
Rational Root Theorem
The Rational Root Theorem provides possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. It helps narrow down candidates to test when searching for roots of polynomials with integer coefficients.
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04:06Rational Exponents
Polynomial Division or Synthetic Division
Polynomial or synthetic division is used to divide a polynomial by a binomial of the form (x - c). It helps simplify the polynomial after finding one root, reducing the degree and making it easier to find remaining zeros.
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07:30Introduction to Factoring Polynomials
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