Question

Find the missing coefficients and exponents identified by question marks. Rewrite the entire equation without question marks when you have solved it.

Answer 1

`(12x^(16)-18x^8-42x^6)/(-6x^5)=-2x^(11)+3x^3+7x`

Explanation:

Replace the questions marks with letters:

`(12x^(a)-18x^8+bx^6)/(cx^d)=ex^(11)+3x^f+7x`

`\textsf{Apply the fraction rule}\quad (a+b+c)/(d)=(a)/(d)+(b)/(d)+(c)/(d)`

`\implies (12x^(a))/(cx^d)-(18x^8)/(cx^d)+(bx^6)/(cx^d)=ex^(11)+3x^f+7x`

Taking the second term and calculating the coefficient of the denominator:

`\implies -(18x^8)/(cx^d)=3x^f`

`\implies (-18)/(c)=3`

`\implies c=(-18)/(3)=-6`

Taking the first term, substituting the found coefficient of the denominator and calculating the coefficient e:

`\implies (12x^(a))/(cx^d)=(12x^(a))/(-6x^d)=ex^(11)`

`\implies e=(12)/(-6)=-2`

Taking the third term, substituting the found coefficient of the denominator, and calculating the coefficient of the numerator and the exponent:

`\implies (bx^6)/(cx^d)=(bx^6)/(-6x^d)=7x`

`\implies 6-d=1\implies d=5`

`\implies (b)/(-6)=7 \implies b=-42`

Taking the first term, substituting the found coefficient and exponent of the denominator and the found coefficient e, and calculating the exponent of the numerator::

`\implies (12x^(a))/(cx^d)=ex^(11)`

`\implies (12x^(a))/(-6x^5)=-2x^(11)`

`\implies a-5=11 \implies a=16`

Taking the second term, substituting the found coefficient and exponent of the denominator, and calculating the exponent f:

`\implies -(18x^8)/(cx^d)=-(18x^8)/(-6x^5)=3x^f`

`\implies f=8-5=3`

Substituting all found letters:

`(12x^(16)-18x^8-42x^6)/(-6x^5)=-2x^(11)+3x^3+7x`