5.6 Summary
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5.5 Solving cubic equations
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5.6 Summary (EMCGY)
Terminology: | |
Expression | A term or group of terms consisting of numbers, variables and the basic operators (\(+, -, \times, \div\)). |
Univariate expression | An expression containing only one variable. |
Root/Zero | A root, also referred to as the “zero”, of an equation is the value of \(x\) such that \(f(x)=0\) is satisfied. |
Polynomial |
An expression that involves one or more variables having different powers and coefficients. \(a_{n}x^{n} + \ldots + a_2x^{2} + a_{1}x + a_{0}, \text{ where } n \in \mathbb{N}_0\) |
Monomial |
A polynomial with one term. For example, \(7a^{2}b \text{ or } 15xyz^{2}\). |
Binomial |
A polynomial that has two terms. For example, \(2x + 5z \text{ or } 26 - g^{2}k\). |
Trinomial |
A polynomial that has three terms. For example, \(a - b + c \text{ or } 4x^2 + 17xy - y^3\). |
Degree/Order |
The degree, also called the order, of a univariate polynomial is the value of the highest exponent in the polynomial. For example, \(7p - 12p^2 + 3p^5 + 8\) has a degree of \(\text{5}\). |
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Quadratic formula: \(x = \frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\)
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Remainder theorem: a polynomial \(p(x)\) divided by \(cx - d\) gives a remainder of \(p\left(\dfrac{d}{c}\right)\).
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Factor theorem: if the polynomial \(p(x)\) is divided by \(cx - d\) and the remainder, \(p \left( \frac{d}{c} \right)\), is equal to zero, then \(cx - d\) is a factor of \(p(x)\).
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Converse of the factor theorem: if \(cx - d\) is a factor of \(p(x)\), then \(p \left( \frac{d}{c} \right) = 0\).
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Synthetic division:
We determine the coefficients of the quotient by calculating:
\begin{align*} q_{2} &= a_{3} + \left( q_{3} \times \frac{d}{c} \right) \\ &= a_{3} \quad \text{ (since } q_{3} = 0) \\ q_{1} &= a_{2} + \left( q_{2} \times \frac{d}{c} \right) \\ q_{0} &= a_{1} + \left( q_{1} \times \frac{d}{c} \right) \\ R &= a_{0} + \left( q_{0} \times \frac{d}{c} \right) \end{align*}
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5.5 Solving cubic equations
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End of chapter exercises
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