How do you divide #(-4x^3-15x^2-4x-12)/(x-4) #?

Question

How do you divide #(-4x^3-15x^2-4x-12)/(x-4)  #?

Isaiah Anthony · Accepted Answer

The remainder is #=-524# and the quotient is #=-4x^2-31x-128#

Now let's execute the artificial division.

#color(white)(aaaa)##4##|##color(white)(aaaa)##-4##color(white)(aaaa)##-15##color(white)(aaaaaa)##-4##color(white)(aaaaa)##-12#

#color(white)(aaaaa)##|##color(white)(aaaa)##color(white)(aaaaaaa)##-16##color(white)(aaaa)##-124##color(white)(aaaa)##-512#

#color(white)(aaaaaaaaa)###_________

#color(white)(aaaaaa)##|##color(white)(aaaa)##-4##color(white)(aaaa)##-31##color(white)(aaaa)##-128##color(white)(aaaa)##color(red)(-524)#

The remainder is #=-524# and the quotient is #=-4x^2-31x-128#

ALSO,

Use the theorem of remainders.

When a polynomial #f(x)# is divided by #(x-c)#, we get

#f(x)=(x-c)q(x)+r#

Let #x=c#

Then,

#f(c)=0+r#

Here,

#f(x)=-4x^3-15x^2-4x-12#

Consequently,

#f(4)=-4*4^3-15*4^2-4*4-12#

#=-256-240-16-12#

#=-524#

The remainder is #=-524#

Julia Adams · Answer

undefined To divide (-4x^3-15x^2-4x-12) by (x-4), you can use long division. Here are the steps:

1. Divide the first term of the numerator (-4x^3) by the first term of the denominator (x). The result is -4x^2.
2. Multiply the entire denominator (x-4) by -4x^2, giving -4x^3 + 16x^2.
3. Subtract this result from the numerator: (-4x^3-15x^2-4x-12) - (-4x^3 + 16x^2) = -31x^2 - 4x - 12.
4. Bring down the next term from the numerator, which is -31x^2. Now you have -31x^2 - 4x - 12.
5. Divide the first term of this new numerator (-31x^2) by the first term of the denominator (x). The result is -31x.
6. Multiply the entire denominator (x-4) by -31x, giving -31x^2 + 124x.
7. Subtract this result from the numerator: (-31x^2 - 4x - 12) - (-31x^2 + 124x) = -128x - 12.
8. Bring down the next term from the numerator, which is -128x. Now you have -128x - 12.
9. Divide the first term of this new numerator (-128x) by the first term of the denominator (x). The result is -128.
10. Multiply the entire denominator (x-4) by -128, giving -128x + 512.
11. Subtract this result from the numerator: (-128x - 12) - (-128x + 512) = -524.
12. There are no more terms left in the numerator, so the division is complete.

The quotient is -4x^2 - 31x - 128, and the remainder is -524.