a) Divide \( x+2 x^{2}+3 x+4 \) by \( x+2 \) b) Divide \( 2 x^{3}-x^{2}+3 x-2 \) by \( x+3 \) c) Divide \( a^{3}+64 b^{3} \) by \( a+4 b \) 2. Find the remainder when \( x^{3}+3 x^{2}-x+7 \) is divided by \( 3 x-1 \).
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The Deep Dive
When dividing polynomials, it’s not just about the numbers; it’s also about the fun process of synthetic or long division! For part a), when you divide \( x + 2x^2 + 3x + 4 \) by \( x + 2 \), you're essentially arranging the terms in descending order, which is like sorting your toys! After simplification, you find the quotient is \( 2x + 3 \) with a remainder of \( 0 \). For part d), the Remainder Theorem comes to the rescue! To find the remainder when dividing \( x^3 + 3x^2 - x + 7 \) by \( 3x - 1 \), you can evaluate the polynomial at \( x = \frac{1}{3} \). Plugging this into the polynomial gives you the remainder, which makes this method not only efficient but a bit like a magic trick in math!