Pascal's Triangle and Binomial Theorem

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Expand the expression:

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Expand the expression:

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Expand the expression:

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Expand the expression:

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Expand the expression:

Open Ended

What patterns can you find after expanding the binomial $$\left(a+b\right)^n$$ , where $$n=1,\ 2,\ 3,\ 4$$ ?

Open Ended

How many terms are there in the expansion of $$\left(a+b\right)^0,\ \left(a+b\right)^1,\ \left(a+b\right)^2,\ \left(a+b\right)^3,\ \ \left(a+b\right)^4$$ ?

Open Ended

How many terms are there in the expansion of $$\left(a+b\right)^n$$ ?

Open Ended

Looking from left to right, what happens to the powers of a? how about the powers of b?

Open Ended

What is the sum of the powers of the variables in each term?

Open Ended

Take the coefficients of each term in each expansion. Predict the coefficients of each term of $$\left(a+b\right)^5$$

Open Ended

Take the coefficients of each term in each expansion. Predict the coefficients of each term of $$\left(a+b\right)^6$$

Multiple Choice

Try this!

Expand: $$\left(a-b\right)^7$$

• $$a^7-7a^6b+21a^5b^2-35a^4b^3+35a^3b^4-21a^2b^5+7ab^6-b^7$$

• $$b^7-7ab^6+21a^2b^5-35a^3b^4+35a^4b^3-21a^5b^2+7a^6b-a^7$$

• $$-a^7+7a^6b-21a^5b^2+35a^4b^3-35a^3b^4+21a^2b^5-7ab^6+b^7$$

• $$-b^7+7ab^6-21a^2b^5+35a^3b^4-35a^4b^3+21a^5b^2-7a^6b+a^7$$

Multiple Choice

Try this!

Expand: $$\left(x+3\right)^5$$

• $$x+3x^2+9x^3+27x^4+9x^5+3$$

• $$x^5+3x^4+9x^3+27x^2+9x+3$$

• $$x^5+15x^4+90x^3+270x^2+405x+243$$

• $$x+15x^2+90x^3+270x^4+405x^5+243$$

Multiple Choice

Try this!

Expand: $$\left(2x-y\right)^4$$

• $$x^4-8x^3y+24x^2y^2-32xy^3+16y^4$$

• $$16x^4-32x^3y+24x^2y^2-8xy^3+y^4$$

• $$-16x^4+32x^3y-24x^2y^2+8xy^3-y^4$$

• $$-x^4+8x^3y-24x^2y^2+32xy^3-16y^4$$