Unit 04 Financial Mathematics (General Mathematics 9 PCTB)
Notes, solutions of unit 04, General Mathematics 9, for Punjab Curriculum & Textbook Board (PCTB), Lahore.
- Know that a rational expression behaves like a rational number.
- Define a rational expression as the quotient $\frac{p(x)} {q(x)}$ of two polynomials $p(x)$ and $q(x)$ where $q(x)$ is not the zero polynomial.
- Examine whether a given algebraic expression is a
- polynomial or not,
- rational expression or not.
- Define $\frac{p(x)} {q(x)}$ as a rational expression in its lowest terms if $p(x)$ and $q(x)$ are polynomials with integral coefficients and having no common factor.
- Examine whether a given rational algebraic expression is in lowest from or not.
- Reduce a given rational expression to its lowest terms.
- Find the sum, difference and product of rational expressions.
- Divide a rational expression with another and express the result in it lowest terms.
- Find value of algebraic expression for some particular real number.
- Know the formulas
- $(a + b)^2 + (a – b)^2 = 2(a^2 + b^2)$,
- $(a + b)^2 – (a – b)^2 = 4ab$
- Find the value of $a^2 + b^2$ and of $ab$ when the values of $a + b$ and $a – b$ are known.
- Know the formulas
- $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.
- Find the value of $a^2 + b^2 + c^2$ when the values of $a + b + c$ and $ab + bc + ca$ are given.
- find the value of $a + b + c$ when the values of $a^2 + b^2 + c^2$ and $ab + bc + ca$ are given.
- find the value of $ab + bc + ca$ when the values of $a^2 + b^2 + c^2$ and $a + b + c$ are given.
- Know the formulas
- $(a + b)^3 = a^3 + 3ab(a + b) + b^3$,
- $(a - b)^3 = a^3 - 3ab(a - b) - b^3$,
- Find the value of $a^3 ± b^3$ when the values of $a ± b$ and $ab$ are given
- Find the value of $x^3 ±$ when the value of $x ±$ is given.
- Know the formulas
- $a^3 ± b^3 = (a ± b)(a^2 ± ab + b^2)$.
- find the product of $x +\frac{1}{x}$ and $x^2 +\frac{1}{x^2}-1$
- find the product of $x-\frac{1} {x}$ and $x^2 +\frac{1}{x^2} +1$
- find the continued product of $(x + y) (x - y) (x^2 + xy + y^2 ) (x^2 - xy + y^2 )$.
- Recognize the surds and their application.
- Explain the surds of second order. Use basic operations on surds of second order to rationalize the denominators and evaluate it.
- Explain rationalization (with precise meaning) of real numbers of the types $\frac{1} { a+ b \sqrt{x}}, \frac{1}{\sqrt{x}+ \sqrt{y}}$ and their combinations where $x$ and $y$ are natural numbers and $a$ and $b$ integers.
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