91Ó°ÊÓ

Why is finding the degree of a polynomial simplified when the polynomial is written in standard form?

a. Show that \((k+1) !=(k+1) \cdot k !\) b. Show that \(n \mathrm{C}_{k}+_{n} \mathrm{C}_{k}+1=n+1 \mathrm{C}_{k+1}\) c. Suppose \(n=4\) and \(k=2 .\) What entries in Pascal's Triangle are represented by \(_{n} \mathrm{C}_{k},_{n} \mathrm{C}_{k}+1,\) and \(_{n}+1 \mathrm{C}_{k}+1\) ? Verify that the equation in part (b) is true for these entries.

Write a polynomial function in standard form with the given zeros. 1, multiplicity 4

Use the discriminant to find the number of real solutions. $$ 3 x^{2}+x-6=0 $$

Simplify each expression. \((-4 i)(6 i)\)

What is the polynomial function, in standard form, whose zeros are \(-2,5,\) and \(6,\) and whose leading coefficient is \(-2 ?\) Justify your reasoning.

What is the expanded form of \((a-b)^{3} ?\) \(\begin{array}{ll}{\text { A. } a^{3}+a^{2} b+a b^{2}+b^{3}} & {\text { B. } a^{3}+3 a^{2} b+3 a b^{2}+b^{3}} \\ {\text { C. } a^{3}-a^{2} b+a b^{2}-b^{3}} & {\text { D. } a^{3}-3 a^{2} b+3 a b^{2}-b^{3}}\end{array}\)

What is the third term in the expansion of \((a-b)^{7} ?\) \(\begin{array}{llll}{\text { F. }-21 a^{5} b^{2}} & {\text { G. }-7 a^{6} b} & {\text { H. } 7 a^{6} b} & {\text { J. } 21 a^{5} b^{2}}\end{array}\)

Use the discriminant to find the number of real solutions. $$ 5 x^{2}-9=0 $$

Simplify each expression. \((2+i)(2-i)\)