4/16 Alg2 Review: Dividing Polynomials

Long Division

100

What is the first step in the process? 

Use long division to divide x³ + x² – 2x + 14 by x + 3.

Divide the leading term of (x+3) into the leading term of x³ + x² – 2x + 14.

200

What will the remainder of the quotient be a fraction of? 

Use long division to divide x³ + x² – 2x + 14 by x + 3.

Fraction of the divisor, x + 3.

300

Use long division to divide 𝘹² − 1𝟥𝘹 - 48 by x+3.

x - 16

400

Use long division to divide x³ + x² – 2x + 14 by x + 3.

x² - 2x + 4 + (2/x+3)

Synthetic Division

100

What is the 'multiplier' value that we will take from the divisor of the following problem: 

Use synthetic division to divide (x³ − 8x² + 17x − 10) ÷ (x − 5).

5

200

What are the coefficients that we will take from the dividend of the following problem: 

Use synthetic division to divide (x³ − 8x² + 17x − 10) ÷ (x − 5).

1, -8, 17, -10

300

If I am left with following values after using synthetic division, what is the quotient?

1, -3, 2, 0

x² - 3x + 2

400

Use synthetic division to divide (x³ − 8x² + 17x − 10) ÷ (x − 5)

x² - 3x + 2

Remainder Theorem & Factor Theorem

100

When P(x) = 2x³ – x² + 4x + 5 is divided by x + 2, the remainder is -23. 

According to the Remainder Theorem, what is the value of P(-2)?

P(-2) = -23

200

When P(x) = 2x³ – x² + 4x + 5 is divided by x + 2, the remainder is -23.

According to the Factor Theorem, is x + 2 a factor of P(x)?

No, because the remainder is not zero. 

300

What is the remainder if f(x) = x⁴ − 5x² − 6x − 10 is divided by x − 3?

f(3) = 8. By the Remainder Theorem, the remainder is 8. 

400

According to the Factor Theorem, how do I know if a binomial (x + a) is a factor of a polynomial P(x)?

If there is a remainder of zero when P(x) is divided by (x+a), then (x+a) is a factor of P(x). 

Is it a factor?

100

See slide 20.

Yes. By the Factor Theorem, since the remainder is zero the binomial is a factor of the polynomial.

200

See slide 21. 

No. By the Factor Theorem, since the remainder is NOT zero the binomial is NOT a factor of the polynomial.

300

What are the two methods of identifying whether or not a binomial, (x - a), is a factor of a polynomial, P(x)?

1. Synthetic Division (According to the work, is the remainder zero?)

2. Direct Substitution (Does P(a)=0?)

400

Is x + 9 a factor of the polynomial 𝘗(𝘹) = 𝘹³ + 𝟣𝟣𝘹² + 𝟣𝟧𝘹 − 𝟤𝟩?                  

Yes. The remainder of the quotient of is 0, so x + 9 is a factor of P(x).