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3) \( (3 x-1) \cdot P(x)=6 x^{3}-5 x^{2}+10 x-3 \)

Ask by Coleman Pierce. in Greece
Jan 21,2025

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\( P(x) = 2x^2 - x + 3 \)

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The Deep Dive

To solve for \( P(x) \), first, we need to express \( P(x) \) in terms of the given equation. Start by dividing both sides by \( (3x - 1) \). This means we will carry out polynomial long division or synthetic division depending on the degree of \( 6x^3 - 5x^2 + 10x - 3 \) versus \( 3x - 1 \). First, arrange the division: 1. Divide the leading term \( 6x^3 \) by \( 3x \) to get \( 2x^2 \). 2. Multiply \( 2x^2 \) by \( (3x-1) \) to obtain \( 6x^3 - 2x^2 \). 3. Subtract this from the original polynomial resulting in: \[ (-5x^2) - (-2x^2) = -3x^2 + 10x - 3 \] 4. Next, divide \( -3x^2 \) by \( 3x \) to get \( -x \). 5. Multiply \( -x \) by \( (3x-1) \) gives \( -3x^2 + x \). 6. Subtract: \[ (10x) - (x) = 9x - 3 \] 7. Now divide \( 9x \) by \( 3x \) to get \( 3 \). 8. Multiply \( 3 \) by \( (3x-1) \) results in \( 9x - 3 \). 9. Subtract: \[ (-3) - (-3) = 0 \] After this division, we conclude that: \[ P(x) = 2x^2 - x + 3 \] This expression shows you the behavior of \( P(x) \) based on the original polynomial's structure!

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