Lesson 24

Polynomial Identities (Part 2)

  • Let’s explore some other identities.

Problem 1

Is \(a^6 + b^6 = (a^2+b^2)(a^4 - a^2b^2 + b^4)\) an identity? Explain or show your reasoning.

Problem 2

Match each lettered expression with the number of an expression equivalent to it.

A:

\(\frac{1}{a} + \frac{1}{a+1}\)

B:

\(\frac{a+1}{a-1} + \frac{a+1}{a}\)

C:

\(\frac{1}{a} + \frac{2}{a+1}\)

D:

\(\frac{a}{a-1} - \frac{1}{a+1}\)

E:

\(\frac{a}{a+1} + \frac{a}{a-1}\)

1:

\(\frac{2a^2}{a^2-1}\)

2:

\(\frac{3a+1}{a^2+a}\)

3:

\(\frac{2a+1}{a^2+a}\)

4:

\(\frac{2a^2+a-1}{a^2-a}\)

5:

\(\frac{a^2+1}{a^2-1}\)

Problem 3

Let \((x^2+5x+4)(x+2)=A(x+1)\). If this is an identity, what is a possible expression for \(A\)?

Problem 4

What are the points of intersection between the graphs of the functions \(f(x)=(x + 6)(2x+1)\) and \(g(x)=2x+1\)?

(From Unit 2, Lesson 11.)

Problem 5

Identify all values of \(x\) that make the equation true.

  1. \(\frac{x+5}{x+11}=\frac{1}{x+5}\)
  2. \(\frac{2x-3}{x} = \frac{14}{x+5}\)
(From Unit 2, Lesson 22.)

Problem 6

Match each expression in the lettered list with the number of an expression equivalent to it.

A:

\((x - 1)(x^3+x^2+x+1)\)

B:

\((x+6)(x-6)\)

C:

\((x-1)^3\)

D:

\(x^4 - 36\)

E:

\((x+2)^3 - x^3\)

1:

\(x^3 - 3x^2 + 3x - 1\)

2:

\((x^2 + 6)(x^2 - 6)\)

3:

\(x^2 - 36\)

4:

\(2(3x^2+6x+4)\)

5:

\(x^4-1\)

(From Unit 2, Lesson 23.)