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[PDF] 32 The Factor Theorem and The Remainder Theorem
The graph suggests that the function has three zeros, one of which is x = 2 It's easy to show that f(2) = 0, but the other two zeros seem to be less
[PDF] AMSG11Remainder and Factor Theorempdf
In this section, we will learn to use the remainder and factor theorems to factorise and to solve polynomials that are of degree higher than 2 Before doing so,
[PDF] POLYNOMIALS - NCERT
Factor Theorem : If p(x) is a polynomial of degree n > 1 and a is any real number, then (i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x –
[PDF] Factor Theorem Examples And Solutions
Class 9 Maths Polynomials Factor Theorem Factor Theorem x a hook a factor of the polynomial px if pa 0 Also if x a passage a factor of px then pa 0 where
[PDF] 9th Class Maths Polynomials-Remainder Theorem - Takshila Learning
Study more about this topic with online classes for 9th Class Maths Remainder Theorem: Let p(x) be any polynomial of degree greater than or equal to one and
[PDF] class:9 polynomials subject: mathematics - dsert
2 sept 2021 · Through examples interpret to find Discuss and recall about definition, CLASS:9 POLYNOMIALS ACTIVITY SHEET-07 FACTOR THEOREM
[PDF] Zeros of a Polynomial Function
3 5 3 5 Factor P x x x x x x x x x x = + + ? = + + ? Page 18 Example 9 (Continued): From the Rational Zeros Theorem (Section 5 3), we obtain the
[PDF] Polynomials Maths Class: 9 By: Manish Gupta
Class: 9 By: Manish Gupta polynomial p(x) thus we define the zero of a polynomial as follows 05 Division of Polynomials and Remainder Theorem
[PDF] Polynomials Class 9 Maths Notes with Formulas Download in pdf
22 avr 2020 · https://ncerthelp com/text php?ques=Polynomials+Class+9+Maths+ degree of zero polynomial is not defined Remainder theorem : Let
[PDF] RS Aggarwal Solutions Class 9 Maths Chapter 2- Polynomials
According to the remainder theorem, p(x) divided by (x-1) obtains the remainder as g(1) Calculating g(1) = 1 3 ? 6(1)2 + 9 × 1 + 3
[PDF] POLYNOMIALS - NCERT
shall also study the Remainder Theorem and Factor Theorem and their use in the role in the collection of all polynomials, as you will see in the higher classes Example 9 : Find the remainder when x4 + x3 – 2x2 + x + 1 is divided by x – 1
[PDF] Definition Mathematics 9th Class Science Punjab Board - MathCityorg
Definitions Mathematics (Science Group): 9th Written by Amir Shahzad, Version: 2 0 Chapter a linear divisor (x-a), then the remainder is f(a) Factor theorem:
[PDF] 32 The Factor Theorem and The Remainder Theorem
The proof of Theorem 3 4 is usually relegated to a course in Abstract Algebra,3 but The Remainder Theorem: Suppose p is a polynomial of degree at least 1 -9 -39 2 -6 -13 -35 2 From this, we get -6x2 - 4x +2 = (x - 3 2)(-6x - 13) - 35 2
[PDF] 6 The factor theorem
linear factors corresponding to the zeros x=1,2 and 4 6 factor theorem 9 2 3 4 + − + − = x x x x xf [This is the polynomial of Example 1 with last term 18 school district was in Taiwan observing an after school class of 400 grade 10
[PDF] Quadratic Equations, the Zero Factor Theorem, and Factoring
the following are all examples of quadratic equations o 2 we will use the Zero Factor Theorem to solve quadratic equations that are in factored 9 2 − 16 = 0 Again, this method of solving equations can be used to solve more than
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101390_6QuadraticEquations,theZeroFactorTheorem,andFactoring.pdf 16-week Lesson 12 (8-week Lesson 10) Quadratic Equations, the Zero Factor Theorem, and Factoring
1Solving quadratic equations will
be the main topic on Exam 3; we will be solving quadratic equations in Lessons 12, 13, 14, and 15. You will still need to understand the (such as rational equations fromLesson 11) and will cover (such as
equations containing fraction exponents from Lesson 15), but solving quadratic equations will make-up about 75% of Exam 3, so please be prepared.16-week Lesson 12 (8-week Lesson 10) Quadratic Equations, the Zero Factor Theorem, and Factoring
2Quadratic equation:
- any equation that can be written in the form ܽݔଶܾݔܿ o ܾ and/or ܿ can equal zero, but ܽ o if ܽ - the degree of a quadratic equation is always - - the following are all examples of quadratic equations o -ݔଶെ͵ݔെͷൌ- o െݔଶͶݔൌ- o ଵ ଶݔଶଷ ସൌ- o െ͵ݔଶൌ- The first method we will use to solve quadratic equations is factoring. If we can take a quadratic equation, which is just a polynomial set equal to zero, and factor that polynomial, we can use the Zero Factor Theorem to solve it.Zero Factor Theorem:
- the product of two or more factors is zero if and only if at least one of the factors is zero o ݔݕൌ- if and only if ݔൌ- or ݕൌ- - we will use the Zero Factor Theorem to solve quadratic equations that are in factored form and set equal to zeroo if ሺݔ-ሻሺݔെ͵ሻൌ-, then ݔ-ൌ- and ݔെ͵ൌ-, which
means ݔൌെ- and ݔൌ͵ - this Theorem does NOT work for any other numbers but zero o if you have an equation in factored form such asሺݔെ-ሻሺݔ͵ሻൌͳ or ሺݔͷሻሺݔെͳሻൌെ͵, we CANNOT
simply set each factor equal to the number to solve; the Zero Factor Theorem only works with zero, because the only way to get a product of zero is to multiply by zero When an equation is factorable, we set it equal to zero so we can use theZero Factor Theorem to solve it.
16-week Lesson 12 (8-week Lesson 10) Quadratic Equations, the Zero Factor Theorem, and Factoring
3Steps for Solving an Equation by Factoring:
1. write the equation as a polynomial and set it equal to zero
2. factor the polynomial (review the Steps for Factoring if needed)
3. use Zero Factor Theorem to solve
Example 1: Solve the quadratic equation ͳͷݔଶെ-ൌͳ͵ݔ for ݔ and enter exact answers only (no decimal approximations). If there is more than one solution, separate your answers with commas. If there are no real solutions, enter BC 3CD4 CB. ͳͷݔଶെ-ൌͳ͵ݔ Quadratic equations are not the only type of equation that can be solved byfactoring. Other polynomial equations such as -ݔସെͳͺݔଶͶͺൌ-
(which we will see in a future lesson) that are not quadratic can still be solved by factoring. ࢇࢉ െ- ࢈ͳ
Think
about the signs of the product and the sum. െͳǡ- െ-ǡ͵- െ͵ǡ-- െͶǡͳͷ െͷǡͳ- െǡͳ-16-week Lesson 12 (8-week Lesson 10) Quadratic Equations, the Zero Factor Theorem, and Factoring
4Example 2: Solve the quadratic equation ݔሺ͵ݔെ-ͷሻൌെͺ for ݔ and
enter exact answers only (no decimal approximations). If there is more than one solution, separate your answers with commas. If there are no real solutions, enter BC 3CD4 CB. ݔሺ͵ݔെ-ͷሻൌെͺ ࢇࢉ െ- ࢈ͳ
Think
about the signs of the product and the sum. െͳǡ- െ-ǡ͵- െ͵ǡ-- െͶǡͳͷ െͷǡͳ- െǡͳ-16-week Lesson 12 (8-week Lesson 10) Quadratic Equations, the Zero Factor Theorem, and Factoring
5 Example 3: Solve the following equations for ݔ and enter exact answers only (no decimal approximations). If there is more than one solution, separate your answers with commas. If there are no real solutions, enterBC 3CD4 CB.
a. ݔൌଵହିସ௫మ ଵ b. -ݔ-Ͷൌͷݔଶ b. b-ൌݔଶെͳݔെͳ- -ൌͷݔଶെ-ݔെ-Ͷ
-ൌݔଶ͵ݔെ--ݔെͳ- -ൌͷݔଶͶݔെ͵-ݔെ-Ͷ
-ൌ͵ݔሺ-ݔͳሻെͳ-ሺ-ݔͳሻ -ൌݔሺͷݔͶሻെሺͷݔͶሻ
-ൌሺ-ݔͳሻሺ͵ݔെͳ-ሻ -ൌሺͷݔͶሻሺݔെሻ
-ൌ-ݔͳ Ǣ -ൌ͵ݔെͳ- ͷݔͶൌ- Ǣ ݔെൌ-
െଵ ଶൌݔ Ǣ ଵ ଷൌݔ ࢞ൌെ Ǣ ࢞ൌ ࢇࢉ െͳ-- ࢈ െ-Think
about the signs of the product and the sum.ͳǡെͳ-- െͳͳͻ
-ǡെ- െͷͺ͵ǡെͶ- െ͵
ǡെ െ16-week Lesson 12 (8-week Lesson 10) Quadratic Equations, the Zero Factor Theorem, and Factoring
6c. ͳ-ͳݔൌݔଶ d. ሺݔെͳሻሺݔെ-ሻൌ
d. d-ൌݔଶെͳݔെͳ- ݔଶെ͵ݔ-ൌ
-ൌݔଶ͵ݔെ--ݔെͳ- ݔଶെ͵ݔെͶൌ-
-ൌ͵ݔሺ-ݔͳሻെͳ-ሺ-ݔͳሻ ݔଶݔെͶݔെͶൌ-
-ൌሺ-ݔͳሻሺ͵ݔെͳ-ሻ ݔሺݔͳሻെͶሺݔͳሻൌ-
-ൌ-ݔͳ Ǣ -ൌ͵ݔെͳ- ሺݔͳሻሺݔെͶሻൌ-
െ ൌ࢞ Ǣ ൌ࢞ ݔͳൌ- Ǣ ݔെͶൌ- െଵ ଶൌݔ Ǣ ଵ ଷൌݔ ࢞ൌെ Ǣ ࢞ൌ e. ݔଶെ-ͷൌ- f. ͻݔଶെͳൌ- Again, this method of solving equations can be used to solve more than just quadratic equations, as we will see in future lessons.Answers to Examples:
1. ݔൌെଶ
ଵହǡͳ ; 2. ݔൌଵ ଷǡͺ ; 3a. ݔൌെͷǡଷ ସ ; 3b. ݔൌെସ ହǡ ;3c. ൌ െଵ
ଶǡଵଷ ; 3d. ݔൌെͳǡͶ ; 3e. ൌെͷǡͷ ; 3f. ݔൌെସ
ଷǡସ ଷ ;