Question

How do you divide `(x^4-5x^3+6x^2-10x+15)/(x^2-7)`?

Answer 1

Perform long division of polynomials.

Explanation:

Remember to write any terms whose coefficient is 0:
`color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) color(white)( (x^4-5x^3+6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)`

Because `x^2*x^2 = x^4`, the first term of the quotient is `x^2`:

`color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2color(white)( 5x^3+6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)`

Multiply the first term in the quotient by the divisor and subtract underneath:

`color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2color(white)( 5x^3+6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)`
`color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr`
`color(white)(.........................)-5x^3+13x^2-10x`

Because `-5x*x^2 =-5x^3`. the next term in the quotient is `-5x`:

`color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2-5xcolor(white)(6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)`
`color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr`
`color(white)(.........................)-5x^3+13x^2-10x`

Multiply the second term in the quotient by the divisor and subtract underneath:

`color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2-5xcolor(white)(6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)`
`color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr`
`color(white)(.........................)-5x^3+13x^2-10x`
`color(white)(..............................)ul(5x^3+color(white)(.)0x^2-45x)" "darr`
`color(white)(........................................)13x^2-45x+15`

Because `13*x^2 =13x^2`. the next term in the quotient is `13`:

`color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2-5x+13color(white)(-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)`
`color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr`
`color(white)(.........................)-5x^3+13x^2-10x`
`color(white)(..............................)ul(5x^3+color(white)(.)0x^2-45x)" "darr`
`color(white)(........................................)13x^2-45x+15`

Multiply the third term in the quotient by the divisor and subtract underneath:

`color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2-5x+13color(white)(-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)`
`color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr`
`color(white)(.........................)-5x^3+13x^2-10x`
`color(white)(..............................)ul(5x^3+color(white)(.)0x^2-35x)" "darr`
`color(white)(........................................)13x^2-45x+15`
`color(white)(.....................................)ul(-13x^2color(white)(.)-0x+91)`
`color(white)(...............................................)-45x+106`

The quotient is `x^2-5x+13` with a remainder of `-45x+10`