 The remainder theorem states that the remainder that you get when you divide a polynomial P(x) by (x – a) is equal to P(a).  The factor theorem is.

Slides:

Advertisements

Similar presentations

Dividing Polynomials Section 2.4. Objectives Divide two polynomials using either long division or synthetic division. Use the Factor Theorem to show that.

Advertisements

Rational Zeros Theorem Upper & Lower Bounds Long Division

Long Division of Polynomials

4.4 Notes The Rational Root Theorem. 4.4 Notes To solve a polynomial equation, begin by getting the equation in standard form set equal to zero. Then.

EXAMPLE 2 Find all zeros of f (x) = x 5 – 4x 4 + 4x x 2 – 13x – 14. SOLUTION STEP 1 Find the rational zeros of f. Because f is a polynomial function.

Pre Calc Lesson 2.2 Synthetic Division ‘Remainder’ and ‘Factor’ Theorems Review Long Division: 5365 ÷ 27 Now review ‘long division’ of polynomials: (2x.

The Remainder and Factor Theorems Check for Understanding 2.3 – Factor polynomials using a variety of methods including the factor theorem, synthetic division,

5.5 Apply the Remainder and Factor Theorem

Sarah Byom and Samantha Kingery. * 1. Divide using synthetic division: X 3 -5x 2 +3x X+1 A. x 2 -6x /x+1B. x 2 -4x+1 C. x 2 +6x+3D. x2-6x+3+-11/x+1.

ACTIVITY 34 Review (Sections ).

Quick Crisp Review Zeros of a polynomial function are where the x-intercepts or solutions when you set the equation equal to zero. Synthetic and long division.

Polynomials Expressions like 3x 4 + 2x 3 – 6x and m 6 – 4m 2 +3 are called polynomials. (5x – 2)(2x+3) is also a polynomial as it can be written.

2.2 D IVIDING POLYNOMIALS ; R EMAINDER AND F ACTOR T HEOREMS.

5.4 – Apply the Remainder and Factor Theorems Divide 247 / / 8.

Polynomial Division and the Remainder Theorem Section 9.4.

1 What we will learn today…  How to divide polynomials and relate the result to the remainder and factor theorems  How to use polynomial division.

7.4 THE REMAINDER & FACTOR THEOREMS Objectives: The student will be able to… 1)evaluate functions using synthetic substitution 2)determine whether a binomial.

2.3 Polynomial Division and Synthetic Division Ex. Long Division What times x equals 6x 3 ? 6x 2 6x x 2 Change the signs and add x x.

ACTIVITY 31: Dividing Polynomials (Section 4.2, pp )

Section 5.3(d) Synthetic Substitution. Long division Synthetic Division can be used to find the value of a function. This process is called Synthetic.

Section 2-2 Synthetic Division; The Remainder and Factor Theorems.

7.3 Products and Factors of Polynomials Objectives: Multiply polynomials, and divide one polynomial by another by using long division and synthetic division.

Factor Theorem Using Long Division, Synthetic Division, & Factoring to Solve Polynomials.

3.6 Day 2 Why Synthetic Division? What use is this method, besides the obvious saving of time and paper?

I CAN USE LONG DIVISION AND SYNTHETIC DIVISION. I CAN APPLY THE FACTOR AND REMAINDER THEOREMS. Lesson 2-3 The Remainder and Factor Theorems.

Using theorems to factor polynomials.  If a polynomial f(x) is divided by x-k, then the remainder r = f(k)  This is saying, when you divide (using synthetic.

Warm Up no 0, 3 x = -3. Homework Questions Section 2.2 Synthetic Division; The Remainder and Factor Theorems Objective: To use synthetic division and.

If a polynomial f(x) is divided by (x-a), the remainder (a constant) is the value of the function when x is equal to a, i.e. f(a). Therefore, we can use.

Theorems About Roots of Polynomial Equations. Find all zeros: f(x)= x +x –x Synthetic Division one zero…need 2 more use (x – k), where.

Remainder Theorem If f(x) is divided by x – r, then the remainder is equal to f(r). We can find f(r) using Synthetic Division.

The Original f(x)=x 3 -9x 2 +6x+16 State the leading coefficient and the last coefficient Record all factors of both coefficients According to the Fundamental.

Dividing Polynomials Section 2.4. Objectives Divide two polynomials using either long division or synthetic division. Use the Factor Theorem to show that.

2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85.

Solving Polynomials. What does it mean to solve an equation?

LESSON 5.6 Rational Zeros of Polynomial Functions.

Quotient Dividend Remainder Divisor Long Division.

Finding Roots With the Remainder Theorem. Dividing Polynomials Remember, when we divide a polynomial by another polynomial, we get a quotient and a remainder.

Section 4.3 Polynomial Division; The Remainder and Factor Theorems Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Sect. 2-2 Synthetic Division; The remainder and Factor theorems Objective: SWBAT use the synthetic division and to apply the remainder and factor theorems.

Long and Synthetic Division. Long Division Polynomial long division can be used to divide a polynomial d(x), producing a quotient polynomial q(x) and.

Solving Polynomials. Factoring Options 1.GCF Factoring (take-out a common term) 2.Sum or Difference of Cubes 3.Factor by Grouping 4.U Substitution 5.Polynomial.

Polynomial Long Division

Remainder Theorem Let f(x) be an nth degree polynomial. If f(x) is divided by x – k, then the remainder is equal to f(k). We can find f(k) using Synthetic.

Section 3.3 Dividing Polynomials; Remainder and Factor Theorems

Dividing Polynomials Section 4.3.

Dividing Polynomials Two options: Long Division Synthetic Division.

Divide x3 + x2 – 10x + 8 by x+4 using long division.

Warm Up Compute the following by using long division.

Section 5.4 – Dividing Polynomials

Divide by x - 1 Synthetic Division: a much faster way!

Dividing Polynomials Long Division A little review:

1a. Divide using long division. (9x3 – 48x2 + 13x + 3) ÷ (x – 5)

2.5 Zeros of Polynomial Functions

5.8 Rational Zero Theorem.

Real Zeros Intro - Chapter 4.2.

Aim #3. 3 How do we divide Polynomials

Finding Roots of Higher Order Polynomials

Section 3.3 Dividing Polynomials; Remainder and Factor Theorems

Apply the Remainder and Factor Theorems

Polynomial Division; The Remainder Theorem and Factor Theorem

6.5 The Remainder and Factor Theorems

Remainder and Factor Theorem

Chapter 2 notes from powerpoints

4.3 – The Remainder and Factor Theorems

8-5 Rational Zero Theorem

Section 2.4: Real Zeros of Polynomial Functions

Section 3.3 Dividing Polynomials; Remainder and Factor Theorems

3.2 The Remainder Theorem.

Presentation transcript:

 The remainder theorem states that the remainder that you get when you divide a polynomial P(x) by (x – a) is equal to P(a).  The factor theorem is a direct consequence of the remainder theorem.

 The factor theorem states that (x – a) is a factor of a polynomial P(x) if and only if P(a) = 0.  This is because if P(a) = 0, the remainder when P(x) is divided by (x – a) is zero as well.

 The factor theorem lets us easily factor polynomials.  If we can find a point a where P(a) = 0, we know that we’ve found a root of P(x) and that we can factor (x-a) out of P(x).  We can do this using synthetic division or long division.

 Find all the roots of f(x)=2x 3 + 3x 2 – 11x – 6.  First, we should find the possible rational roots of the function. Using the rational roots test, we find that they are ±1, ±2, ±3, ±6, ±1/2, and ±3/2.  Let’s start testing points to see if we can find a root.

xf(x) f(2) = 0, so (x-2) is a factor of our polynomial. Let’s divide f(x) by x-2 using synthetic division to begin factoring it.

 This tells us that we can factor f(x) as (x – 2)(2x 2 + 7x + 3).  We can factor this quadratic comparatively easily. Our final result is (x – 2)(2x+1)(x+3).  If we didn’t have a quadratic, we could use synthetic division again on our quotient.  The roots of our function are 2, -1/2. and -3.