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6 The factor theorem
6 The factor theorem www mast queensu ca/~peter/investigations/6factors pdf It's worth pointing out that cubic equations are not so easy to solve If the equation in Example 3 were quadratic, we could use the quadratic formula, but it's
23 Factor and remainder theorems
2 3 Factor and remainder theorems www surrey ac uk/sites/default/files/2021-07/2 3-factor-and-remainder-theorems pdf knowledge and skills in working with the factor and remainder theorems It 2 3 3 Apply the remainder theorem Example: Remainder theorem
32 The Factor Theorem and The Remainder Theorem
3 2 The Factor Theorem and The Remainder Theorem www shsu edu/~kws006/Precalculus/2 3_Zeroes_of_Polynomials_files/S 26Z 203 2 pdf Example 3 2 1 Use synthetic division to perform the following polynomial divisions Find the quotient and the remainder polynomials, then write the dividend,
AMSG11Remainder and Factor Theorempdf
AMSG 11 Remainder and Factor Theorem pdf irp-cdn multiscreensite com/f15f3f52/files/uploaded/AMSG 11 Remainder 20and 20Factor 20Theorem pdf For example, we may solve for x in the following equation as follows: Hence, x = ?3 or ?2 are solutions or roots of the quadratic equation A more general
22 - The Factor Theorem
2 2 - The Factor Theorem vanvelzermath weebly com/uploads/2/3/5/2/23525212/2 2_the_factor_theorem pdf 24 fév 2015 Use long division to determine the other factors Page 6 6 February 24, 2015 Example Five Factor fully
Factor Theorem
Factor Theorem mr-choi weebly com/uploads/1/7/0/5/17051620/2-2_-_factor_theorem pdf 2 2 - Factor Theorem Factor Theorem Example 1a: Use the factor theorem to determine which binomials are factors of the polynomial
428 - The Factor Theorem - Scoilnet
4 2 8 - The Factor Theorem - Scoilnet www scoilnet ie/uploads/resources/28744/28480 pdf Example 1 Q Suppose f (x)=5x3 - 14x2 + 12x - 3 (i) Is (x - 2) a factor? (ii) Is (x - 1) a factor? 4 2 - Algebra - Solving Equations 4 2 8 - The Factor
Factor Theorem - jongarvincom
Factor Theorem - jongarvin com jongarvin com/up/MHF4U/slides/factor_theorem_handout pdf Factor Theorem J Garvin Slide 1/14 polynomial equations & inequalities Factor Theorem Example Divide f (x) = x3 + 4x2 + x - 6 by x - 1
The Factor Theorem and a corollary of the - UMass Blogs
The Factor Theorem and a corollary of the - UMass Blogs blogs umass edu/math421-murray/files/2010/08/FactorTheoremEvaluated pdf 27 août 2010 the latter inequality says that the remainder r is less than the “divisor” b For example, if you use long division to divide 2356 by 14, you
L3 – 22 – Factor Theorem Lesson MHF4U - jensenmath
L3 – 2 2 – Factor Theorem Lesson MHF4U - jensenmath www jensenmath ca/s/22-ls-factor-theorem pdf a) Use the remainder theorem to determine the remainder when Example 1: Determine if ?3 and +2 are factors of ( ) = ? ? 14 + 24
51 The Remainder and Factor Theorems; Synthetic Division
5 1 The Remainder and Factor Theorems; Synthetic Division users math msu edu/users/bellro/mth103fa13/mth103fa13_chapter5 pdf use the factor theorem Example 1: Use long division to find the quotient and the remainder: 27 5593 ÷ Steps for Long Division:
99584_622_ls_factor_theorem.pdf L3 - 2.2 - Factor Theorem Lesson
MHF4U
Jensen
In this section, you will learn how to determine the factors of a polynomial function of degree 3 or
greater.Part 1: Remainder Theorem Refresher
a) Use the remainder theorem to determine the remainder when ( ) = ! +4 " +-6 is divided by +2b) Verify your answer to part a) by completing the division using long division or synthetic division.
Factor Theorem:
- is a factor of a polynomial () if and only if ( ) =0. Similarly, - is a factor of () if and only if # $ 0=0.Remainder Theorem: When a polynomial function
() is divided by -, the remainder is (); and when it is divided by -, the remainder is ! " ) , where and are integers, and ≠0.Note: I chose synthetic since it is
a linear divisor of the form -. Example 1: Determine if -3 and +2 are factors of ( ) = ! - " -14+24 ( 3 ) =Since the remainder is ___, -3 divides evenly into (); that means -3 _______________________ of ().
( -2 ) =Since the remainder is not ____, +2 does not divide evenly into (); that means +2
____________________________ of (). Part 2: How to determine a factor of a Polynomial With Leading Coefficient 1 You could guess and check values of that make ( ) =0 until you find one that works... Or you can use the Integral Zero Theorem to help.Integral Zero Theorem
If - is a factor of a polynomial function () with leading coefficient 1 and remaining coefficients that
are integers, then is a factor of the constant term of ().Note: Once one of the factors of a polynomial is found, division is used to determine the other factors.
Example 2: Factor
! +2 " -5-6 fully.Let
( ) = ! +2 " -5-6Find a value of such that
( ) =0. Based on the factor theorem, if ( ) =0, then we know that - is a factor. We can then divide () by that factor. The integral zero theorem tells us to test factors of _____Test ________________________________. Once one factor is found, you can stop testing and use that factor to
divide (). ( 1 ) = Since ________________, we know that ________________________ a factor of (). ( 2 ) = Since ________________, we know that ____________________ a factor of (). You can now use either long division or synthetic division to find the other factors Method 1: Long division Method 2: Synthetic DivisionExample 3: Factor
% +3 ! -7 " -27-18 completely.Let
( ) = % +3 ! -7 " -27-18Find a value of such that
( ) =0. Based on the factor theorem, if ( ) =0, then we know that - is a factor. We can then divide () by that factor. The integral zero theorem tells us to test factors of ________Test ________________________________________________. Once one factor is found, you can stop testing and use that
factor to divide ().Since ________________, this tell us that ____________ is a factor. Use division to determine the other factor.
We can now further divide
! +2 " -9-18 using division again or by factoring by grouping.Method 1: Division
Method 2: Factoring by Grouping
Therefore,
% +3 ! -7 " -27-18=Example 4: Try Factoring by Grouping Again
% -6 ! +2 " -12Note: Factoring by grouping
does not always work...but when it does, it saves you time! Group the first 2 terms and the last 2 terms and separate with an addition sign.Common factor within each group
Factor out the common binomial
Part 3: How to determine a factor of a Polynomial With Leading Coefficient NOT 1The integral zero theorem can be extended to include polynomials with leading coefficients that are not 1.
This extension is known as the rational zero theorem.Rational Zero Theorem:
Suppose
( ) is a polynomial function with integer coefficients and = # $ is a zero of () , where and are integers and ≠0. Then, • is a factor of the constant term of () • is a factor of the leading coefficient of () • (-) is a factor of ()Example 5: Factor
( ) =3 ! +2 " -7+2We must start by finding a value of
# $ where # $ 0=0. must be a factor of the constant term. Possible values for are: ____________ must be a factor of the leading coefficient. Possible values of are: ____________Therefore, possible values for
# $ are: ________________________________Test values of
# $ for in () to find a zero.Since ________________________________________________ of (). Use division to find the other factors.
Example 6: Factor
( ) =2 ! + " -7-6Part 4: Application Question
Example 7: When
( ) =2 ! - " +-2 is divided by +1, the remainder is -12 and -2 is a factor. Determine the values of and .Hint: Use the information given to create 2 equations and then use substitution or elimination to solve.