Features: Standard Form vs Factored Form of a Polynomial

Factored Form

Standard Form

Features Observed in Standard Form vs Factored Form of a Polynomial

Standard Form

You can easily identify:

• Degree (highest exponent)

• Leading coefficient (coefficient of the highest-degree term)

• End behavior (using degree + leading coefficient)

• y-intercept (constant term 𝑐)

• Overall shape of the graph (general behavior)

Standard Form

Harder to see:

• Exact x-intercepts (zeros)

• Multiplicity of zeros

Factored Form

You can easily identify:

• x-intercepts (zeros)

• Multiplicity of each zero

• Whether the graph crosses or touches the x-axis at each zero

• Degree (by adding the exponents on the factors)

Factored Form

Harder to see:

• y-Intercept

• Expanded equation

• Precise end behavior without checking the leading coefficient

Multiple Choice

What are the x-intercepts?

• 0 and 8

• -2 and -4

• 3 and -1

• 2 and 4

Multiple Choice

Using the degree and leading coefficient, match the graph to an equation.

• y=(2x-1)(x+2)²
• y=(2x-1)(x+2)³
• y=(-2x-1)(x+2)²
• y=(-2x-1)(x+2)

Multiple Choice

Which description best matches the function shown:

• Odd degree with positive leading coefficient.

• Even degree with positive leading coefficient.

• Degree three, with negative leading coefficient.

• Degree four, with negative leading coefficient.

Multiple Choice

Which function has this end behavior?

As x gets larger and larger in either positive or negative direction, y gets larger and larger in the positive direction.
Note: f(x)=y

• -x²-3x+1
• -x³+2x²+3
• x⁴+3x³-4x+1
• x⁵-4x⁴+2x²-1

Multiple Choice

Based on the end behavior of the polynomial, what best describes the leading coefficient?

• a negative real number
•  a positive real number
• an even real number
• an odd real number

Multiple Choice

The end behavior of a polynomial function is determined by the degree and the sign of the leading coefficient.
Identify the degree of the polynomial and the sign of the leading coefficient 

• Leading Coefficient Positive
  Degree - Even
• Leading Coefficient Positive
  Degree - Odd
• Leading Coefficient Negative
  Degree - Even
• Leading Coefficient Negative
  Degree - Odd

Dropdown

Determine the zeros of $$f\left(x\right)=\left(2x^2-x-1\right)\left(x^2-3x-4\right)$$ . To determine the zeros, factor each quadratic trinomial.

Your result should be f(x)=​ ​​

Select answer

​

Select answer

​

Select answer

​ ​

Select answer

Multiple Choice

A function with degree 3 is called

• constant

• cubic

• linear

• quadratic

Multiple Choice

What is the degree of the polynomial function
P(x)=3x⁴-7x²-2x⁷-x+4?

• 4
• 7
• 2
• 1

Multiple Choice

What is the type of polynomial 3x⁴ + 2x² + 5x ?

• monomial
• binomial
• trinomial
• polynomial

Multiple Choice

What is the standard form of the polynomial,
 9x² + 5x + 27 + 2x³

• 27 + 5x + 9x² + 2x³
• 9x² + 5x + 2x³ + 27
• 9x² + 5x + 27 + 2x³
• 2x³ + 9x² + 5x + 27

Multiple Choice

What is the leading coefficient?

• Positive 
• Negative

Multiple Choice

What is the end behavior for
f(x)=2x²+x

• A
• B
• C
• D

Multiple Choice

Is (x-7) a factor of (x²-5x+14)?

• Yes
• No

Multiple Choice

Find the y-intercept for the polynomial : $$y=5x^3-6x^2+3x-5$$

• -6.28

• -5

• 0

• 0.8