# If f(x)=x^3+Ax^2+Bx-3 and if f(1)=4 andf(-1)=-6, what is the value of 2A+B? (A) 12 (B) 8 (C) 0 (D) -2

Elianna Lawrence 2022-07-26 Answered
If $f\left(x\right)={x}^{3}+A{x}^{2}+Bx-3$ and if f(1)=4 and f(-1)=-6, what is the value of 2A+B?
(A) 12
(B) 8
(C) 0
(D) -2
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Marisa Colon
$f\left(x\right)={x}^{3}+A{x}^{2}+Bx-3$ and if f(1)=4 and f(-1)=-6, what is the value of 2A+B?
$f\left(1\right)=4=1+A+B-3⇒A+B=6$.....(I)
$f\left(-1\right)=-6=-1+A-B-3⇒A-B=-2$......(II)
2A =4
$A=2⇒A+B=6⇒B=4$
Hence 2A+B= 4+4=8
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Shannon Andrews
We have that f(1)=4 and that f(-1)=6. This gives us:
$f\left(1\right)=4\left(1{\right)}^{3}+A\left({1}^{2}\right)+B\left(1\right)-3=A+B-2$
$f\left(-1\right)=-6=\left(-1{\right)}^{3}+A\left(-{1}^{2}\right)+B\left(-1\right)-3=A-B-4$
So, we have:
4=A+B-2....(1)
-6=A-B-4.....(2)
Adding (1) and (2) gives us:
4+(-6)=(A+B-2)+(A-B-4)
-2=2A-6=> 2A=4
Now, we subtract both of them and get:
4-(-6)=(A+B-2)-(A-B-4)
10=2B+2 => B=4
Therefore, 2A+B = 4+4 = 8 , which is option (B)
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