Polynomial division
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Polynomial division works similarly to ordinary long division, with the divisor polynomial having a lower degree than the dividend polynomial. The dividend is written as a multiple of the divisor plus a remainder, where the degree of the remainder is less than the degree of the original dividend. This process is repeated with the remainders as new dividends until the final remainder's degree is less than the divisor's degree. As an example, the document shows the polynomial 3x^3 + 2x^2 - 7x + 5 being divided by x + 3 through repeated use of this process.
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This document discusses binomial expansion, which is a method for expanding binomials like (x + a)^n without lengthy multiplication. It introduces key concepts like Pascal's triangle for finding coefficients and the binomial theorem for determining the general pattern of terms in the expansion. Examples are worked through to demonstrate expanding specific binomials like (2x - 3y)^6 according to this method.
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The document defines an equation as a condition of equality between two mathematical expressions. It provides examples of how to frame a simple equation using variables, coefficients, and constants. The key properties of equations are also outlined, such as how the left and right sides can be interchanged or terms can be added/subtracted/multiplied/divided if done to both sides. Examples are given to demonstrate solving equations using transposition or addition/subtraction/multiplication. Finally, word problems are presented and solved to find unknown values using simple equations.
P1-Chp13-Integration.pptx
1. The document discusses integration and areas under curves. It provides examples of indefinite integration by finding antiderivatives given derivative functions.
2. Definite integration allows finding the exact area under a curve between limits, unlike indefinite integration which has a constant of integration. Examples are worked through of evaluating definite integrals.
3. Problem solving with definite integrals is demonstrated, such as finding the possible values of a constant P given the value of a definite integral involving P.
Equivalent equations
The document discusses equations and how to solve them. It defines an equation as a mathematical statement indicating two expressions are equal. There are two types: numerical equations with numbers and algebraic equations with variables. The goal in solving equations is to find the value of the variable by rewriting the equation in progressively simpler equivalent forms until the variable is isolated on one side. To do this, the same operation must be applied to both sides of the equation so equivalence is maintained.
English math dictionary
english mathematics dictionary
kamus bahassa inggris untuk matematika
oleh neneng
Nurwaningsih
(06081281520066)
Nurwaningsih30@gmail.com
PROGRAM STUDI PENDIDIKAN MATEMATIKA
FAKULTAS KEGURUAN DAN ILMU PENDIDIKAN
UNIVERSITAS SRIWIJAYA
INDRALAYA
2017
semoga bermanfaat
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Polynomial division
- 1. Polynomial Division Polynomial division works like ordinary division with remainder. For example; 14 ÷ 4 can be written out as follows: 14 = 3 × 4 + 2 The idea is to “take out” as many 4s as we can until its no longer possible. Here is the calculation written out fully: 14 = 1 × 4 + 10 10 = 1 × 4 + 6 6 = 1 × 4 + 2 4 can be taken out 3 times. Each time we take out a 4, check the remainder to see if another 4 can be taken out. 2 is the final remainder since there are no more 4s to pull out. 14 = 3 × 4 + 2 Polynomial division works in exactly this way. Before going through a full example, we’ll see how to take one polynomial out of another.
- 2. The following are all examples of polynomials: 푃1 푥 = 3푥7 − 5푥4 + 3푥3 − 5푥2 + 푥 − 9 푃2 푥 = 6푥3 − 8푥2 + 푥 − 9 푃3 푥 = 푥10 푃4 푥 = 2푥2 − 푥 + 4 The degree of the polynomial is the highest power of 푥 in the polynomial: deg 푃1 푥 = 7 deg 푃2 푥 = 3 deg 푃3 푥 = 10 deg 푃4 푥 = 2 푃5 푥 = 24 deg 푃5 푥 = 0 Constants can be regarded as polynomials of degree 0. Linear functions are polynomials of degree 1. Quadratics are polynomials of degree 2. Cubics are polynomials of degree 3. The Leading Term of a polynomial is the term with the highest power of 푥 as highlighted above. With ordinary division with remainder, the number you divide by is smaller than the number you divide into. With polynomial division, the polynomial you divide by has smaller degree than the polynomial you divide into.
- 3. Polynomial division (see next page) is simply repeated use of the following simple trick: If 푃(푥) and 푄(푥) are two polynomials with deg 푄 푥 < deg 푃 푥 , then we can always write 푃(푥) as a multiple of 푄(푥) with a remainder: 푃 푥 = 푚 푥 푄 푥 + 푟(푥) with deg 푟 푥 < deg 푃 푥 . Here’s an example to illustrate: 푃 푥 = 3푥3 + 2푥2 − 7푥 + 5 푄 푥 = 푥 + 3 푚(푥) is simply what is needed so that the leading terms match on both sides: 3푥3 + 2푥2 − 7푥 + 5 = 푚 푥 × 푥 + 3 + 푟(푥) 3푥3 = 푚 푥 × 푥 ⇒ 푚 푥 = 3푥2 3푥3 + 2푥2 − 7푥 + 5 = 3푥2 푥 + 3 + 푟(푥) r(푥) is simply the difference between 푃(푥) and 푚 푥 푄(푥) 푟 푥 = 3푥3 + 2푥2 − 7푥 + 5 − 3푥2 푥 + 3 = −7푥2 − 7푥 + 5 Finally: 3푥3 + 2푥2 − 7푥 + 5 = 3푥2 푥 + 3 − 7푥2 − 7푥 + 5.
- 4. Example (Polynomial Division) (3푥3 + 2푥2 − 7푥 + 5) ÷ (푥 + 3) Write 3푥3 + 2푥2 − 7푥 + 5 as a multiple of 푥 + 3 with remainder 3푥2 (푥 + 3) + 푟1 Write 푟1 as a multiple of 푥 + 3 with remainder −7푥 (푥 + 3) + 푟2 Write 푟2 as a multiple of 푥 + 3 with remainder + 푟3 14 (푥 + 3) 3푥3 + 2푥2 − 7푥 + 5 = Subtract across to find the remainder: 푟1 = 3푥3 + 2푥2 − 7푥 + 5 − 3푥2(푥 + 3) 푟1 = −7푥2 − 7푥 + 5 −7푥2 − 7푥 + 5 = 푟2 = −7푥2 − 7푥 + 5 + 7푥(푥 + 3) 푟2 = 14푥 + 5 14푥 + 5 = 푟3 = 14푥 + 5 − 14(푥 + 3) 푟3 = −37 We stop now since the degree of the remainder is less than the degree of the divisor. 3푥3 + 2푥2 − 7푥 + 5 We now put this all back together: = 3푥2 (푥 + 3) + 푟1 = 3푥2 (푥 + 3)−7푥(푥 + 3) + 푟2 = 3푥2 (푥 + 3) −7푥(푥 + 3) + 14(푥 + 3) + 푟3 Finally, factorise out (푥 + 3) = 3푥2 푥 + 3 − 7푥 푥 + 3 + 14 푥 + 3 − 37 = (푥 + 3) (3푥2 −7푥 + 14) − 37
- 5. Here’s the same calculation set out as a long division: 3푥2 − 7푥 + 14 푥 + 3 3푥3 + 2푥2 − 7푥 + 5 3푥3 + 9푥2 −7푥2 − 7푥 + 5 −7푥2 − 21푥 14푥 + 5 14푥 + 42 −37 3푥3 푥 3푥2 × (푥 + 3) −7푥 × (푥 + 3) −7푥2 푥 14푥 푥 14 × (푥 + 3) subtract to find remainder subtract to find remainder subtract to find remainder 3푥3 + 2푥2 − 7푥 + 5 = 3푥2 − 7푥 + 14 푥 + 3 − 37