Question

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$f(x) = -2x$
$g(x) = 8x^2 - 5x + 7$
find $(f \cdot g)(x)$.
\\(\circ\\) $-16x^2 + 10x - 14x$
\\(\circ\\) $-16x^3 + 10x^2 - 14x$
\\(\circ\\) $-16x^3 - 5x + 7$
\\(\circ\\) $-16x^4 + 10x^3 - 14x^2$

Explanation:

Step1: Recall the definition of function multiplication

To find \((f\cdot g)(x)\), we need to multiply the two functions \(f(x)\) and \(g(x)\), so \((f\cdot g)(x)=f(x)\cdot g(x)\). Given \(f(x) = - 2x\) and \(g(x)=8x^{2}-5x + 7\), we substitute these into the formula: \((f\cdot g)(x)=(-2x)\cdot(8x^{2}-5x + 7)\).

Step2: Distribute \(-2x\) across the terms in \(g(x)\)

Using the distributive property \(a(b + c + d)=ab+ac + ad\), where \(a=-2x\), \(b = 8x^{2}\), \(c=-5x\), and \(d = 7\):

• Multiply \(-2x\) and \(8x^{2}\): \((-2x)\cdot(8x^{2})=-16x^{3}\)
• Multiply \(-2x\) and \(-5x\): \((-2x)\cdot(-5x)=10x^{2}\)
• Multiply \(-2x\) and \(7\): \((-2x)\cdot7=-14x\)

Step3: Combine the results

Combine the three products we got in Step 2: \((f\cdot g)(x)=-16x^{3}+10x^{2}-14x\)

Answer:

\(-16x^{3}+10x^{2}-14x\) (corresponding to the second option: \(\boldsymbol{-16x^{3}+10x^{2}-14x}\))