Lesson 6

Different Forms

  • Let’s use the different forms of polynomials to learn about them.

Problem 1

\(f(x)=(x+3)(x-4)\) and \(g(x)=\frac13(x+3)(x-4)\). The graphs of each are shown here.

Two parabolas on a coordinate plane.
  1. Which graph represents which polynomial function? Explain how you know.

Problem 2

For each polynomial function, rewrite the polynomial in standard form. Then state its degree and constant term.

  1. \(f(x)=(x+1)(x+3)(x-4)\)
  2. \(g(x)=3(x+1)(x+3)(x-4)\)

Problem 3

Tyler incorrectly says that the constant term of \((x + 4)(x - 4)\) is zero.

  1. What is the correct constant term?
  2. What is Tyler’s mistake? Explain your reasoning.

Problem 4

Which of these standard form equations is equivalent to \((x+1)(x-2)(x+4)(3x+7)\)?

A:

\(x^4 + 10x^3 + 15x^2 - 50x - 56\)

B:

\(x^4 + 10x^3 + 15x^2 - 50x + 56\)

C:

\(3x^4 + 16x^3 + 3x^2 - 66x - 56\)

D:

\(3x^4 + 16x^3 + 3x^2 - 66x + 56\)

Problem 5

Select all polynomial expressions that are equivalent to \(5x^3 +7x - 4x^2 + 5\).

A:

\(13x^{5}\)

B:

\(5x^3 - 4x^2 + 7x + 5\)

C:

\(5x^3 + 4x \boldcdot 2 + 7x + 5\)

D:

\(5 + 4x - 7x^2 + 5x^3\)

E:

\(5 + 7x - 4x^2 + 5x^3\)

(From Unit 2, Lesson 2.)

Problem 6

Select all the points which are relative minimums of this graph of a polynomial function.

points A, B, C, D, E, F, G graphed on a function. A and G are on increasing portions. C is on a decreasing portion. relative minimums at D and F. relative maximums at B and E.
A:

Point \(A\)

B:

Point \(B\)

C:

Point \(C\)

D:

Point \(D\)

E:

Point \(E\)

F:

Point \(F\)

G:

Point \(G\)

(From Unit 2, Lesson 3.)

Problem 7

What are the \(x\)-intercepts of the graph of \(y=(3x+8)(5x-3)(x-1)\)?

(From Unit 2, Lesson 5.)