Given f(x)=3x3+8x2+27x+72
is a polynomial with f(−83)=0
, factor f(x)
completely.(1 point)
Responses
f(x)=(3x+8)(x+3)(x−3)
f left parenthesis x right parenthesis equals left parenthesis 3 x plus 8 right parenthesis left parenthesis x plus 3 right parenthesis left parenthesis x minus 3 right parenthesis
f(x)=3(x−83)(x2+9)
f left parenthesis x right parenthesis equals 3 left parenthesis x minus Start Fraction 8 over 3 End Fraction right parenthesis left parenthesis x squared plus 9 right parenthesis
f(x)=(x−83)(3x2+16x+2093)
f left parenthesis x right parenthesis equals left parenthesis x minus Start Fraction 8 over 3 End Fraction right parenthesis left parenthesis 3 x squared plus 16 x plus Start Fraction 209 over 3 End Fraction right parenthesis
f(x)=(3x+8)(x2+9)
f(x)=(3x+8)(x+3)(x−3)
f(x)=(3x+8)(x+3)(x-3)
To find the factors of the polynomial f(x), we first need to determine the roots of the equation f(x) = 0. In this case, we are given that f(-83) = 0, so -83 is a root of the polynomial.
To factor the polynomial completely, we can divide it by (x + 83) using polynomial long division or synthetic division. The quotient will be a quadratic expression.
Using polynomial long division, the steps would be as follows:
______
x + 83 | 3x^3 + 8x^2 + 27x + 72
- (3x^3 + 249x^2)
_______________
-241x^2 + 27x
- (-241x^2 - 19903x)
_______________
19930x + 72
- (19930x + 1667590)
_______________
-1667520
The remainder is -1667520, which means that the given polynomial is divisible by (x + 83). Therefore, one of the factors is (x + 83).
Next, we can divide the quotient obtained from the previous step, which is -241x^2 + 27x, by (x - 3) to find the remaining factors. Again, using polynomial long division, the steps would be as follows:
______
x - 3 | -241x^2 + 27x
- (-241x^2 - 723x)
_______________
750x
- (750x - 2250)
_______________
2250
The remainder is 2250, which means that the quotient -241x^2 + 27x can be further factored as x - 3. Therefore, the remaining factors are (x - 3) and (x + 83).
Finally, we have factored the polynomial completely as follows:
f(x) = (x + 83)(x - 3)(-241x^2 + 27x)
or f(x) = (x + 83)(x - 3)(3x^2 - 1x)