Algebra II - Algebraic Manipulations

7 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Polynomial Long Division

   Write the long division sign with numerator inside and the denominator outside Divide the firm term of the numerator (inside) by finding what the highest term of the denominator (outside) needs to be multiplied by to yield it, and write that number above Multiple the first term of the numerator (inside) by the entire denominator (outside) and subtract that polynomial Perform the same style of division on this difference and repeat until the degree of the denominator (outside) can no longer divi…

2. Synthetic Division (For a monic linear denominator, ex. x−ax-ax−a )

   Write the synthetic division sign, with the coefficients of the terms of numerator in degree order on the inside, and the zero of the denominator on the outside Drop the first coefficient below the synthetic division sign Multiply this number by the outside number and write it below the next term in the numerator Add the just completed column, and write the sum under the synthetic division sign Repeat steps 3 and 4 with these column sums until there are no more coefficients to operate on Using…

3. Partial fractions

   Rational functions in lowest terms, ex. p(x)q(x)\frac{p(x)}{q(x)}q(x)p(x)​ , where the degree of q(x)q(x)q(x) is greater than that of p(x)p(x)p(x) , can be factored into simpler rational functions, called partial fractions. These can be trivially integrated using natural logarithms.

4. Partial fraction decomposition when the denominator contains distinct linear factors (Heaviside cover-up method)

   If the denominator contains distinct linear factors, ex. c(x+a)(x+b)\frac{c}{(x+a)(x+b)}(x+a)(x+b)c​ , then the fraction can be decomposed by Setting up fractions in the form A(x+a)+B(x+b),...\frac{A}{(x+a)}+\frac{B}{(x+b)},...(x+a)A​+(x+b)B​,... for each linear factor of the denominator Determine the coefficients of each of these partial fractions In the original fraction, mentally "cover-up" or remove the denominator's linear factor associated with the desired coefficient, ex. (x+a)(x+a)(x+a)…

5. Partial fraction decomposition when the denominator contains a repeated linear factor

   If the denominator contains repeated linear factors, ex. c(x+a)n\frac{c}{(x+a)^n}(x+a)nc​ , then the decomposition of the fraction includes the terms A1(x+a)+A2(x+a)2...An(x+a)n\frac{A_1}{(x+a)}+\frac{A_2}{(x+a)^2}...\frac{A_n}{(x+a)^n}(x+a)A1​​+(x+a)2A2​​...(x+a)nAn​​ , and can be decomposed by Setting up partial fractions as normal (each term of the denominator) in addition to the rule above, ex. ` cx(x+a)n=Ax+B(x+a)+C(x+a)2...D(x+a)n\frac{c}{x(x+a)^n}=\frac{A}{x}+\frac{B}{(x+a)}+\frac{C}{(x+…

6. Partial fraction decomposition when the denominator contains a distinct irreducible quadratic factor (cannot be factored into linear factors)

   If the denominator contains a distinct irreducible quadratic factor ax2+bx+cax^2 + b x + cax2+bx+c , then the decomposition of the fraction includes the term Ax+Bax2+bx+c\frac{Ax+B}{ax^2+bx+c}ax2+bx+cAx+B​ , and can be decomposed by Setting up partial fractions as normal (each term of the denominator) in addition to the rule above, ex. 3xx3−1=3x(x−1)(x2+x+1)=A(x−1)+Bx+Cx2+x+1\frac{3x}{x^3-1}=\frac{3x}{(x-1)(x^2+x+1)}=\frac{A}{(x-1)}+\frac{Bx+C}{x^2+x+1}x3−13x​=(x−1)(x2+x+1)3x​=(x−1)A​+x2+x+1Bx+…

7. Partial fraction decomposition when the denominator contains a repeated irreducible quadratic factor (cannot be factored into linear factors)

   If the denominator contains a repeated irreducible quadratic factor (ax2+bx+c)n(ax^2+bx+c)^n(ax2+bx+c)n , then the decomposition of the fraction includes the terms A1x+B1(ax2+bx+c)+A2x+B2(ax2+bx+c)2+Anx+Bn(ax2+bx+c)n\frac{A_1x+B_1}{(ax^2+bx+c)}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\frac{A_nx+B_n}{(ax^2+bx+c)^n}(ax2+bx+c)A1​x+B1​​+(ax2+bx+c)2A2​x+B2​​+(ax2+bx+c)nAn​x+Bn​​ , and can be decomposed by Setting up partial fractions as normal (each term of the denominator) in addition to the rule above, ex.…