Question 1088858
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If x+2 is a factor of x^3 - ax- 6,then find the remainder when 2x^3+ax^2-6x+9 is divided by x+1.
Please help me with this question. 
Thank You.
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<U>Let me do it in a way in how it SHOULD BE DONE</U>.


<pre>
1.  If (x+2) is a factor of {{{x^3 -ax - 6}}}, then, according to the <U>Remainder theorem</U>, the number -2 is the root of the polynomial p(x) = {{{x^3 -ax - 6}}}.

    In other words, p(-2) = 0,  which means

    {{{(-2)^3-a*(-2) - 6}}} = 0,   or, simplifying,

    -8 + 2a - 6 = 0,   or   2a = 8 + 6 = 14  ====>  a = {{{14/2}}} = 7.



3.  Hence, the second polynomial is 

    q(x) = {{{2x^3 + 7x^2 - 6x + 9}}}.


    Then,  again, due to the <U>Remainder theorem</U>, the remainder of division q(x) by (x+1) is equal to the value q(-1), i.e.

     q(-1) = {{{2*(-1)^3 + 7*(-1)^2 - 6*(-1) + 9}}} = 2*(-1) + 7*1 + 6 + 9 = 20.


<U>Answer</U>. The remainder of division the second polynomial by (x+1) is 20.
</pre>

Solved.



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The remainder theorem:

<pre>
    <B>1</B>. The remainder of division the polynomial &nbsp;{{{f(x)}}}&nbsp; by the binomial &nbsp;{{{x-a}}}&nbsp; is equal to the value &nbsp;{{{f(a)}}}&nbsp; of the polynomial. 

    <B>2</B>. The binomial &nbsp;{{{x-a}}}&nbsp; divides the polynomial &nbsp;{{{f(x)}}}&nbsp; if and only if the value of &nbsp;{{{a}}}&nbsp; is the root of the polynomial &nbsp;{{{f(x)}}}, &nbsp;i.e. &nbsp;{{{f(a) = 0}}}.

    <B>3</B>. The binomial &nbsp;{{{x-a}}}&nbsp; factors the polynomial &nbsp;{{{f(x)}}}&nbsp; if and only if the value of &nbsp;{{{a}}}&nbsp; is the root of the polynomial &nbsp;{{{f(x)}}}, &nbsp;i.e. &nbsp;{{{f(a) = 0}}}.
</pre>


See the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Divisibility-of-polynomial-f%28x%29-by-binomial-x-a.lesson>Divisibility of polynomial f(x) by binomial (x-a) and the Remainder theorem</A> 

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic 
"<U>Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem</U>".



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Ignore writing by @josgarithmetic.  His way <U>IS NOT</U> the method for solving such problems.


It is not his level of knowledge and is not his area of expertise.