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[PDF] 32 The Factor Theorem and The Remainder Theorem
The Remainder Theorem: Suppose p is a polynomial of degree at least 1 The proof of The Factor Theorem is a consequence of what we already know
[PDF] 6 The factor theorem
If f(x) is a polynomial and f(a) = 0, then (x–a) is a factor of f(x) Proof of the factor theorem Let's start with an example Consider 4 8 5
[PDF] 23 Factor and remainder theorems
Irreducible quadratic: a quadratic expression which cannot be written as a product of two linear factors using the set of real numbers, R, e g , 2 + 1 = 0 In
[PDF] Factor Theorem Examples And Solutions
Factor theorem state with proof examples and solutions factorise the Polynomials Maths Mutt Solution Here feel some examples of using the Factor Theorem
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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] 51 The Remainder and Factor Theorems; Synthetic Division
(Refer to page 506 in your textbook for more examples ) Example 5: Use both long and short (synthetic) division to find the quotient and remainder for the
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q is a factor of the leading coefficient n a Example 1: List all possible rational zeros given by the Rational Zeros Theorem of P(x) = 6x
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[PDF] Quadratic Equations, the Zero Factor Theorem, and Factoring
the following are all examples of quadratic equations o 2 2 ? 3 ? 5 = 0 we will use the Zero Factor Theorem to solve quadratic equations
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The Remainder Theorem for divisor ( − ) From the above examples, we saw that a polynomial can be expressed as a product of the quotient and the
[PDF] Quadratic Equations, the Zero Factor Theorem, and Factoring
the following are all examples of quadratic equations o 2 2 − 3 − 5 zero, and factor that polynomial, we can use the Zero Factor Theorem to solve it
[PDF] 1 31 THE REMAINDER THEOREM AND THE FACTOR - CSUN
1 3 1 THE REMAINDER THEOREM AND THE FACTOR THEOREM Definition: Example 1 Is 0 a zero of ( ) 1 2 3 + − = xx xP I DIVISION OF POLYNOMIALS
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101394_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 ;