Polynomial and identities in practice

The following exercises are solved in this step.

We invite you to try to solve them before watching the video.

In any case, you will find below a PDF file with the solutions.

Exercise 1.

Verify the following identity for (ngeq 3):

[(a-b)(a^{n-1}+a^{n-2}b+cdots+ab^{n-2}+b^{n-1})=a^n-b^n.]

Deduce that if (n) is odd then

[(a+b)(a^{n-1}-a^{n-2}b+cdots+(-1)^{n-2}ab^{n-2}+(-1)^{n-1}b^{n-1})=] [=a^n+b^n,]

and if (n) is even then

[(a+b)(a^{n-1}-a^{n-2}b+cdots+(-1)^{n-2}ab^{n-2}+(-1)^{n-1}b^{n-1})=] [=a^n-b^n.]

Exercise 2. [Solved only in the PDF file]

Decompose the following polynomials using the notable identities:

i) (9x^2-12x+4)

ii) (x^4+2x^2y+y^2)

iii) (9x^2-1)

iv) (3x^2-y^4)

v) (8x^3+27y^3)

vi) (frac{1}{8}x^6-1)

Exercise 3. [Solved only in the PDF file]

Simplify the following expressions:

i) ((a^m+a^n)^2 – (a^m +a^n)(a^m-a^n)-2a^n(a^m+a^n))

ii) ((x-y+frac{1}{2})(x-y-frac{1}{2})-(x-y)^2+(frac{1}{3}x^2-frac{3}{2})^3+)

(-x^2(frac{1}{27}x^4-frac{1}{2}x^2+frac{9}{4}))

iii) (frac{1}{2}(2x+frac{1}{2})^2-2(2x-frac{1}{2})^2)