Polynomial Evaluation and Long Division

This video tutorial teaches how to evaluate polynomials for a given value using substitution and long division. It begins with a basic example of evaluating a polynomial by substituting a value for x and simplifying using the order of operations. The tutorial then explains the process of long division with polynomials, highlighting common misunderstandings such as sign changes in binomials. The remainder theorem is introduced, demonstrating its application in polynomial division. An advanced example is provided to reinforce the concepts, showing both substitution and long division methods.

What is the first step in evaluating a polynomial for a given value?

• Apply long division

• Use the remainder theorem

• Substitute the given value into the polynomial

• Factor the polynomial

When substituting x = 3 into the polynomial x^2 + 2x - 7, what is the result?

• 12

• 8

• 5

• 10

What is the first step in long division of polynomials?

• Subtract the remainder

• Add placeholders for missing terms

• Multiply the divisor by the quotient

• Place the numerator under the division sign

What is a common mistake when forming a binomial from a solution?

• Forgetting to multiply

• Not changing the sign

• Adding instead of subtracting

• Using the wrong divisor

What does the remainder theorem state?

• The remainder is always zero

• The remainder is the value of the polynomial at a given point

• The remainder is the divisor

• The remainder is the quotient

How do you verify a polynomial evaluation using long division?

• By factoring the polynomial

• By graphing the polynomial

• By checking the remainder

• By using the quadratic formula

What is the result of evaluating 2x^3 - 7 for x = -3 using substitution?

• -61

• 27

• -27

• 61

Why is it important to add zeros as placeholders in long division?

• To avoid negative remainders

• To make the division easier

• To ensure all terms are accounted for

• To simplify the polynomial

What is the remainder when dividing 2x^3 - 7 by x + 3?

• 61

• -7

• 0

• -61

What is the final step in polynomial long division?

• Bring down the next term

• Subtract the last product from the dividend

• Multiply the divisor by the last term

• Add the quotient and remainder