Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.

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Accepted Answer

Welcome back. I'm so glad you're here. We are given this polynomial 18 x cubed plus 74 X squared minus X plus K. And we're told a couple of things about it. We're told that nine X plus one is a factor of that polynomial and we are asked to solve for K. The first thing I want to do is recall from previous lessons. The factor theorem. The factor theorem has a couple parts but I want to focus on the part that tells us that if X minus C is a factor of F of X, then F of C equals zero. So here we're told that nine X plus one is a factor of this polynomial. So I'm going to name this polynomial F of X. And then we're told that if we put C in for X F of C, we're going to get This polynomial equal to zero for F of C. So we need to just figure out what C is. Let's kind of zoom in over here. If X minus C equals zero because x minus C is a factor, then nine X plus one equals zero. To solve for C. Let's add suitable sides and then we get X equals C. Now let's solve for X in er nine X plus one equals zero. We'll subtract one from both sides and we get nine X equals negative one Will divide both sides by nine and then we'll get X equals negative 1/9 or C equals negative 1/9. And now we can plug this C in to our polynomial. So anywhere there was an X. Will now plug in our negative 1/9. So it'll be 18 times not X. But negative 1/9 cubed plus 74 times not X squared but negative 1/9 squared minus 145 times not X but negative 1/9 and scoop that over plus K. And now the only variable we're dealing with is K. So we can solve for K. Let's do the algebra. Starting with the exponents negative 1/9 cubed. That is negative one 729th. So we've got negative one 729th times plus 74 times negative 1/9 squared is one 81st positive. One 81st minus 145 times negative 1/9 plus K equals zero. Now to do the multiplication 18 times negative. One 729th is negative. 2 80 firsts plus 74 times one 81st is 74 80 firsts minus 145 times negative. 1/9. Well that's going to be a positive. So plus 145. 9th plus K equals zero. Now adding these three fractions on a calculator that is going to give us 17. So we have 17 plus K equals zero solving for K. Let's subtract 17 from both sides and we now have K equals negative 17 and that's it. We look at our answer choices in this matches with answer choice B. Well done, we'll catch you on the next one.

Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.

Key Concepts

Polynomial Factorization

Remainder Theorem

Substitution Method