Answer:
[tex]\dfrac{12x^{16}-18x^8-42x^6}{-6x^5}=-2x^{11}+3x^3+7x[/tex]
Step-by-step explanation:
Replace the questions marks with letters:
[tex]\dfrac{12x^{a}-18x^8+bx^6}{cx^d}=ex^{11}+3x^f+7x[/tex]
[tex]\textsf{Apply the fraction rule}\quad \dfrac{a+b+c}{d}=\dfrac{a}{d}+\dfrac{b}{d}+\dfrac{c}{d}[/tex]
[tex]\implies \dfrac{12x^{a}}{cx^d}-\dfrac{18x^8}{cx^d}+\dfrac{bx^6}{cx^d}=ex^{11}+3x^f+7x[/tex]
Taking the second term and calculating the coefficient of the denominator:
[tex]\implies -\dfrac{18x^8}{cx^d}=3x^f[/tex]
[tex]\implies \dfrac{-18}{c}=3[/tex]
[tex]\implies c=\dfrac{-18}{3}=-6[/tex]
Taking the first term, substituting the found coefficient of the denominator and calculating the coefficient e:
[tex]\implies \dfrac{12x^{a}}{cx^d}=\dfrac{12x^{a}}{-6x^d}=ex^{11}[/tex]
[tex]\implies e=\dfrac{12}{-6}=-2[/tex]
Taking the third term, substituting the found coefficient of the denominator, and calculating the coefficient of the numerator and the exponent:
[tex]\implies \dfrac{bx^6}{cx^d}=\dfrac{bx^6}{-6x^d}=7x[/tex]
[tex]\implies 6-d=1\implies d=5[/tex]
[tex]\implies \dfrac{b}{-6}=7 \implies b=-42[/tex]
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::
[tex]\implies \dfrac{12x^{a}}{cx^d}=ex^{11}[/tex]
[tex]\implies \dfrac{12x^{a}}{-6x^5}=-2x^{11}[/tex]
[tex]\implies a-5=11 \implies a=16[/tex]
Taking the second term, substituting the found coefficient and exponent of the denominator, and calculating the exponent f:
[tex]\implies -\dfrac{18x^8}{cx^d}=-\dfrac{18x^8}{-6x^5}=3x^f[/tex]
[tex]\implies f=8-5=3[/tex]
Substituting all found letters:
[tex]\dfrac{12x^{16}-18x^8-42x^6}{-6x^5}=-2x^{11}+3x^3+7x[/tex]
