Polynomials Remainder Theorem

Polynomial f(x) Review

Reviewing Polynomial Functions, Roots, Division, and Graphical Behavior

Necessary Skills

• The Factor Theorem: Understanding that if f(a) = 0, then (x - a) is a factor of the polynomial function f(x)

• The Remainder Theorem: When using division with a particular divisor, if the remainder is 0, then the divisor is a factor of the dividend polynomial function p(x)

• End Behavior: Based on degree (highest exponent) as well as leading coefficient (positive or negative)

Multiple Choice

Even or Odd degree?

• even

• odd

Multiple Choice

What degree is the function?

• 1
• 2
• 4
• 10

Multiple Choice

Which function is graphed?

• (x - 4)(x + 2)(x - 8)=y

• (x - 4)(x - 2)(x + 8)=y

• (x - 4)(x + 2)(x + 8)=y

• (x + 4)(x + 2)(x + 8)=y

Multiple Choice

Is the leading coefficient positive or negative

• positive
• negative

Multiple Choice

f(x) = (x+2)(x-3)²(x-4)

Which choice best describes the zeros?

• -2, 3 (multiplicity 2), 4

• -2 (multiplicity 2), 3, 4

• 2, -3 (multiplicity 2), -4

• 2 (multiplicity 2), -3, -4

Multiple Choice

What is the degree of the function graphed here?

• 6
• 4
• 3
• 5

Multiple Choice

Which of the following could be the equation of this graph in factored form? (Careful-pay attention to multiplicity.)

• f(x) = (x-4)(x-1)²(x+2)(x+4)

• f(x) = (x-4)(x+1)²(x+2)(x+4)

• f(x) = (x+4)(x+1)(x-2)(x+4)

• f(x) = (x+4)(x+1)²(x-2)(x-4)

Multiple Choice

If a zero of a polynomial function has multiplicity 4 that means:

• The x intercept is at (4, 0)
• The graph will "bounce back" at that zero
• The graph will "pass through" at that zero
• Zeros cannot have multiplicity of 4

Multiple Choice

Write a 4th degree polynomial function with:
*a positive leading coefficient
*zeros: 9, 3, -2, 0

• y = x(x - 9)(x - 3)(x + 2)
• y = x(x + 9)(x + 3)(x - 2)
• y = -x(x - 9)(x - 3)(x + 2)
• y = -x(x + 9)(x + 3)(x - 2)

Multiple Choice

• y =  ^_(x + 2)(x - 1)²
• y = -(x - 2)(x + 1)²
• y = (x - 2)(x + 1)²
• y = -x³

How do we use Division?

• When given a polynomial p(x), we may also be given a potential factor (x - a). We divide by (x - a) to determine if it is a factor...if the remainder = 0, success.

• We can then analyze the quotient to see if it is factorable and will reveal additional factors and eventually roots/zeros of the function p(x)

Multiple Choice

Find the zeros of the polynomial given one factor.

• X= 3,4,5
• x=-3,-4,-5
• x=-3,4,5
• x= 3, -4, -5

Multiple Choice

Write the polynomial in standard form given the following zeros. x = -2, 1, 4

• f(x) = (x+2)(x-1)(x-4)

• f(x) =x³ - 3x² - 6x + 8

• f(x) = x³ + 8

• f(x) = x³ - 3x² + 8

Multiple Choice

How could you determine    if x-2 is a factor of 2x³-5x²+x-2?

• Use synthetic division and see if the quotient is even
• Ask the person sitting next to me
• Use synthetic division and see if the remainder is zero
• Flip a coin

Multiple Choice

What is the Degree of this polynomial:
f(x)= x³(x + 3)²(x – 5)

• Degree: 6
• Degree: 5
• Degree: 3
• Degree: 2

Multiple Choice

Is (x-4) a factor of (x³ +x² -16x-16)?

• YES
• NO

Multiple Choice

What are the
zeros of the polynomial
y = 3(x + 4)(x + 1)(x - 3)?

• -4, -1, 3
• 3, -3, 4, 1
• 4, 1, -3
• 3, -4, -1, 3

Multiple Choice

Multiply the following together:

(2x + 3)(x² + 4x + 1)

• 2x³ + 8x² +2x

• 3x² + 12x +3

• 2x³ +11x² +14x +3

• 2x³ + 12x +3

Multiple Choice

Factor this expression:

2x³ + 2x² - 24x

(Hint: GCF and Slide-n-Divide)

• 2x(x + 4)(x - 3)

• 2x(x - 4)(x - 3)

• 2x(x - 4)(x + 3)

• none of these

Multiple Choice

If (x + 3) divides a polynomial function evenly, then which of these is known to be a zero of the polynomial?

• x = -3

• x = 3

• x = 0

• cannot be determined

Multiple Choice

When a polynomial function f(x) is divided by
 (x + 2), the remainder is 0. 
What is f(-2)?

• -2
• 0
• 2
• cannot be determined

Multiple Choice

Factor this difference of cubes:
x³ - 343

• (x + 7) (x² - 7x + 49)
• (x² + 1) (x - 343)
• (x - 7) (x² + 7x + 49)
• (x² - 343) (x + 1)

Multiple Choice

Factor this sum of cubes:
2x³ + 128

• 2(x³ + 64)
• (x + 4)(x² - 4x + 16)
• 2(x - 4)(x² + 4x + 16)
• 2(x + 4)(x² - 4x + 16)