Mastery Check Solutions

Show What You Know

How can Pascal’s Triangle be used with the Binomial Theorem?

Sample: You can use Pascal’s Triangle to determine the coefficient of the term before simplifying, instead of using combinations.

Note

The $n^{t h}$ term $= {(r + 1)}^{t h}$ term.

Explain what happens to the exponents of each term as you work through the Binomial Theorem.

Sample: The first term’s exponent starts with n and decreases until it reaches zero. The second term’s exponent starts with zero and increases until it reaches the $n^{t h}$ power.

Determine the thirteenth term of the binomial: ${(0.5 x + y)}^{14}$

$\begin{matrix} r = 13 - 1 = 12 \\ n C r (14, 12) = 91 \\ 91 {(0.5 x)}^{2} {(y)}^{12} & n = 14 \\ 91 (0.25) x^{2} y^{12} & 14 - 12 = 2 \\ 22.75 x^{2} y^{12} \end{matrix}$

Note

You can use Pascal’s Triangle handout or the combination formula.

Q: How can you determine the exponents without writing all the terms?

A: The exponent of the second term is $r - 1$ . The first exponent plus the second exponent must equal n.

Determine the middle term of the binomial: ${(0.5 x + y)}^{14}$

$Middle term = 8 th term r = 7 n C r (14, 7) = 3, 432 3, 432 {(0.5 x)}^{7} {(y)}^{7} 3, 432 {(0.5)}^{7} \cdot x^{7} y^{7} 26.8125 x^{7} y^{7}$

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Apply Pascal’s Triangle to problems.
Know and apply the Binomial Theorem to binomial expressions.