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math 1111 - 08 > assignments > homework assignment # 6
homework assignment # 6
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current learning objective: using polynomial division to solve application problems
question 4
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the price of n books is $-4x^{4}+32x^{3}+8x^{2}-66x + 16$. if the price of 6 books is $6x - 48$, express n in terms of x.
$n=\frac{-4x^{4}+32x^{3}+8x^{2}-66x + 16}{x - 8}$
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Explanation:

Step1: Polynomial long - division setup

We want to divide the polynomial \(P(x)=-4x^{4}+32x^{3}+8x^{2}-66x + 16\) by \(D(x)=x - 8\).
The first term of the dividend \(P(x)\) is \(-4x^{4}\) and the first term of the divisor \(D(x)\) is \(x\). The first term of the quotient \(Q(x)\) is \(\frac{-4x^{4}}{x}=-4x^{3}\).

Step2: Multiply and subtract

Multiply \(D(x)\) by \(-4x^{3}\): \((x - 8)\times(-4x^{3})=-4x^{4}+32x^{3}\).
Subtract this from \(P(x)\): \((-4x^{4}+32x^{3}+8x^{2}-66x + 16)-(-4x^{4}+32x^{3})=8x^{2}-66x + 16\).

Step3: Repeat the process

The first term of the new dividend \(8x^{2}-66x + 16\) is \(8x^{2}\). Dividing by \(x\) (from \(D(x)\)), we get the next term of the quotient \(8x\).
Multiply \(D(x)\) by \(8x\): \((x - 8)\times8x=8x^{2}-64x\).
Subtract: \((8x^{2}-66x + 16)-(8x^{2}-64x)=-2x + 16\).

Step4: One more term

The first term of \(-2x + 16\) is \(-2x\). Dividing by \(x\) gives \(-2\).
Multiply \(D(x)\) by \(-2\): \((x - 8)\times(-2)=-2x + 16\).
Subtract: \((-2x + 16)-(-2x + 16)=0 + 736\) (using the remainder formula \(P(x)=Q(x)D(x)+R\), where \(R = 736\)).

So, \(\frac{-4x^{4}+32x^{3}+8x^{2}-66x + 16}{x - 8}=-4x^{3}+8x^{2}+72x + 90+\frac{736}{x - 8}\).

Answer:

\(-4x^{3}+8x^{2}+72x + 90+\frac{736}{x - 8}\)