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- 2. • Al- Khwarizmiis the Father of Algebra. • Abu Ja’far Muhammad ibn Musa al-Khwarizmi lived in Baghdad around 780 to 850 CE (or AD). • The word algebra comes from his book “Kitab Muhtasar fi Hisab Al Gabr Wa I Muqabala” later translataed to English (The Compendious Book of Calculation by Completion and Balancing) • Algebra was described as the science of ‘REDUCTION’ and ‘BALANCING’ Algebra
- 3. 1. Variables Terms Relatedto Algebra • These are symbols or letters that may take one or more than one value. • These variables may represent elements of the set of real numbers.
- 4. 2. Constants Terms Relatedto Algebra • These are numbers or symbol which has a fixed numerical value. • For example: 2, 5, 0, -3, -7, etc. are constants.
- 5. 3. Terms Terms Relatedto Algebra • Term is a constant (number), a variable (letter) or combination of both by a product or by a quotient of constants and variables. • Terms are separated by the symbols plus or minus sign.
- 6. 3. Terms Terms Relatedto Algebra • Term is a constant (number), a variable (letter) or combination of both by a product or by a quotient of constants and variables. • Terms are separated by the symbols plus or minus sign.
- 7. Terms Related to Algebra •Numerical coefficient – the constant factor in a given algebraic expression. The number appears before a variable in a term. o In the expression -5x, the numerical coefficient is -5. o In the expression 10𝑎3 𝑏, the numerical coefficient is 10. • Literal coefficient – the variable including the exponent in a term. o In the expression 9𝑥2 , the literal coefficient is 𝑥2 . 3. Terms
- 8. Terms Related to Algebra SimilarTerms – terms having the same literal coefficients. o 𝟗𝒙𝟐 and 𝟑𝒙𝟐 are similar because their literal coefficients are the same. o 𝟓𝒂𝟐 and 𝒂𝟑 are not similar because their literal coefficients are not the same. o 𝟐𝒙𝒚𝟐 and −𝟑𝒙𝟐𝐲 are not similar because their literal coefficients are not the same. 3. Terms
- 9. Terms Related to Algebra 4.Algebraic Expressions These are mathematical phrases that may made up of variables, constants, or combination of both which may be related by any of the four fundamental operations. Note: it is only a phrase, not a sentence, therefore it does not include relationship sign such as =, <, >, ≤, ≥ and others.
- 10. Terms Related to Algebra 5.Degree the highest exponent or the highest sum of exponents of the variables in a term.
- 11. Terms Related to Algebra 6.Leading Term The term with the highest degree. 7. Leading Coefficient The numerical coefficient of the leading term.
- 12. Algebraic expression Variable/ s Constant/s Numerical Coefficient s Literal Coefficien ts Number of terms Degree Leadin g term Leading Coefficie nt -7 2𝑎 2𝑎𝑏3 𝑥+ 5 𝑛2 −𝑝2 - 4 1 2 𝑥 + 5 8m𝑛2 𝑝2 - 4 5𝑝3 -2p + 5 𝑥2 2 + 5xy − 5
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- 14. What is aPolynomial? Polynomials Polynomial is a special type of algebraic expression in which each term is a constant, or a variable, or a constant multiplied by a positive integral power of one or more variables.
- 15. Example of Polynomials: Polynomials 5 𝑥+ 2 2𝑥2 − 3𝑥 + 5 4𝑥2 − 3𝑥𝑦 + 5𝑦 16𝑎2 − 9𝑏2 2𝑥 + 𝑦 3 𝑥 3 1 2 𝑥3 − 3
- 16. Example of Non-polynomials: Polynomials 3𝑥 2𝑥2 − 3𝑥 3 2 + 5 4𝑥2 − 3𝑥𝑦 + 5𝑦−3 2 + 𝑦 𝑥 2 𝑥 − 𝑦
- 17. When does analgebraic expression be a polynomial? Polynomials Must not have rational/fractional exponent. Must not have negative exponent. Must not have variable/s in the denominator. Must not have variable/s under the radical sign.
- 18. Classification on Polynomials Polynomials Thereare 2 ways of classifying or naming a polynomial these are: • according to number of terms • according to the Degree
- 19. Polynomials according tonumber of terms: Polynomials Monomial – a polynomial with only one term. Binomial – a polynomial with exactly two terms. Trinomial – a polynomial with exactly three terms. Multinomial – a polynomial with four or more terms.
- 20. Polynomials according todegree: Polynomials Constant – a polynomial with degree 0. Linear – a polynomial with degree 1. Quadratic – a polynomial with degree 2. Cubic – a polynomial with degree 3. Quartic – a polynomial with degree 4 Quintic – a polynomial with degree 5 𝒏𝒕𝒉 degree polynomial – use to describe polynomials with degree 6 and more. (𝑛 is the degree of the polynomial)
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- 22. The four mathematicaloperation: Polynomials Addition Subtraction Multiplication Division
- 23. Addition and Subtractionof Polynomials Polynomials • Addition and Subtraction of Polynomials is very simple: 1. Combine the similar terms. 2. Under similar terms just add/subtract its numerical coefficients. 3. In case there are terms dissimilar to the other just add it to the result (copy).
- 24. A review onaddition and subtraction of integers: Polynomials • Addition of Integers remember the: SADS “Same Add, Different Subtract”
- 25. Polynomials • Addition ofIntegers “Same Add, Different Subtract” • Same sign add then follow the sign of the number. • Different sign, Subtract then follow the sign of the greater number.
- 26. Polynomials • Subtraction ofIntegers remember the: 3C-SADS “Copy – Change – Change back to SADS”
- 27. Polynomials • Subtraction ofIntegers “Copy – Change – Change back to SADS” • Copy the first number • Change the operation to addition • Change the sign of the second number • Back to the addition rule.
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- 30. Multiplication and Divisionof Polynomials Polynomials • Addition and Subtraction of Polynomials is very simple: 1. Distributive Property. 2. Divide the numerical coefficients 3. Apply laws of exponents on the literal coefficients.
- 31. A review onmultiplying and dividing integers: Polynomials • In multiplying and Dividing remember the: LPUN “Like Positive, Unlike Negative”
- 32. Polynomials • multiplying andDividing Integers “Like Positive, Unlike Negative” • Sign of the Product and/or Quotient of like sign (same sign) always positive. • Sign of the Product and/or Quotient of unlike sign (different sign) always negative.
- 33. A review onLaws of Exponents: Polynomials • In multiplying and Dividing remember the: LPUN “Like Positive, Unlike Negative”