Given that \(\displaystyle{y}={3}{x}^{{3}}+{7}{x}^{{2}}-{48}{x}+{49}\) and that y has

atenienseec1p

atenienseec1p

Answered question

2022-03-27

Given that y=3x3+7x248x+49 and that y has the same remainder when it is divided by x+k or xk, find the possible values of k.

Answer & Explanation

clarkchica44klt

clarkchica44klt

Beginner2022-03-28Added 17 answers

Step 1
f(k)=f(k)
3(k)3+7(k)248(k)+49=3(k)3+7(k)248(k)+49
bf3k37k2+48k+49=3k3+7k248k+49
3k33k37k27k2+48k+48k+4949=0
6k3+14k2+96k=0
6k3k+14k2k+96kk=0k
6k2+14k96=0
Right there. 7(k)2=7k2.
Looking through it, there are more sign errors throughout.
7k27k2
would be 14k2, not 14k2
Another one: 3k33k3=0
Dividing by k isn't exactly a kosher operation. What if k=0
Chad Robinson

Chad Robinson

Beginner2022-03-29Added 6 answers

Step 1
By the factor theorem,
3x3+7x248x+49=(ax+b)(x2k2)+c
In other words, dividing by a quadratic term leaves a linear quotient and a constant remainder, for the powers on both sides to match.
Observe that there is only one way to make the terms of x3,x2,x
Thus ax2=3x3x=3 and hence
ak2=48k2=16, k=±4

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