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5.6 Summary (EMCGY)

Terminology:
ExpressionA term or group of terms consisting of numbers, variables and the basic operators (\(+, -, \times, \div\)).
Univariate expressionAn expression containing only one variable.
Root/ZeroA 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 \(\text{5}\).

  • Quadratic formula: \(x = \frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\)

  • Remainder theorem: a polynomial \(p(x)\) divided by \(cx - d\) gives a remainder of \(p\left(\dfrac{d}{c}\right)\).

  • 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 \(p(x)\).

  • Converse of the factor theorem: if \(cx - d\) is a factor of \(p(x)\), then \(p \left( \frac{d}{c} \right) = 0\).

  • Synthetic division:

    90dce9977541474c65c21a1e6c2ca835.png

    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*}