Property 9
\(x^n-y^n\quad\text{and}\quad x^n+y^n\)
factoring expressions in the form of
- \(\text{(n is a positive integer)}\)
\(x^n-y^n=\left(x-y\right)\left(x^{n-1}+x^{n-2}\cdot y+x^{n-3}\cdot y^2+\ldots+x\cdot y^{n-2}+y^{n-1}\right)\)
In each term the sum of the exponents of x and y is n - 1.
In each term · the exponent of x is decreased by 1 while the exponent of y is increased by 1.
- \(\text{(n-is a positive odd integer)}\)
\(x^n+y^n=\left(x+y\right)\left(x^{n-1}-x^{n-2}\cdot y+x^{n-3}y^2-\ldots-xy^{n-2}+y^{n-1}\right)\)
1.
\(\left(2a+3b\right)\cdot\left(16a^4-24a^3b+36a^2b^2-54ab^3+81b^4\right)=211\)
\(b=1\)
\(\Rightarrow a=?\)
2.
\(x=\sqrt[7]2\)
\(\Rightarrow\left(x-2\right)\cdot\left(x^6+2x^5+4x^4+8x^3+16x^2+32x+64\right)=?\)
3.
\(\samefrac{a^3+27}{a^2-2a-3}\cdot\samefrac{\left(a-3\right)\cdot\left(a^2-1\right)}{a^2-3a+9}=?\)
4.
\(\samefrac{x^5-1}{x^4+x^3+x^2+x+1}\cdot\samefrac{2x+2}{x^2-1}=?\)
5.
\(x^4+x^3+x^2+x=9\)
\(\Rightarrow x^5-10x=?\)
6.
\(x^4-2x^3+4x^2-8x+16=8\)
\(\Rightarrow x^5-8x=?\)
7.
\(x^4+x^3+x^2+x+1=0\)
\(\Rightarrow x^{10}+x^5=?\)
8.
\(x^4-3x^3+9x^2-27x+81=0\)
\(\Rightarrow x=?\)
9.
\(x=\samefrac45\)
\(\Rightarrow\samefrac{x^{62}+x^{61}+x^{60}+\ldots+1}{1-x^{63}}=?\)
10.
\(\samefrac{5a^4+5a^3+5a^2+5a+5}{a^5-1}:\samefrac{10a+10}{a^2-1}=?\)
11.
\(\left(x+y\right)\left(x^4-x^3y+x^2y^2-xy^3+y^4\right)=244\)
\(y=1\)
\(\Rightarrow x=?\)
12.
\(\left(a-b\right)\left(a^4+a^3b+a^2b^2+ab^3+b^4\right)=31\)
\(b=1\)
\(\Rightarrow a=?\)
13.
\(x=\sqrt[5]7\)
\(\Rightarrow\left(x-1\right)\left(x^4+x^3+x^2+x+1\right)=?\)