Pregunta
Find the remainder when \( 3 x^{2}-2 x+4 \) divided by \( x+2 \).
Ask by Martin Patel. in Nigeria
Feb 22,2025
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The remainder is 20.
Solución
To find the remainder when the polynomial 3x² - 2x + 4 is divided by x + 2, we can use the Remainder Theorem. The theorem states that the remainder of the division of a polynomial f(x) by (x - a) is equal to f(a).
Since our divisor is x + 2, we can rewrite it as (x - (-2)). Therefore, we set a = -2 and evaluate the polynomial at x = -2:
f(-2) = 3(-2)² - 2(-2) + 4
Now, calculate the value step by step:
(-2)² = 4
3 × 4 = 12
-2(-2) = 4
So, f(-2) = 12 + 4 + 4 = 20
Thus, the remainder when dividing 3x² - 2x + 4 by x + 2 is 20.
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To find the remainder when dividing \( 3x^{2} - 2x + 4 \) by \( x + 2 \), we can use the Remainder Theorem. According to this theorem, the remainder of a polynomial \( f(x) \) when divided by \( x - c \) is simply \( f(c) \). In this case, we want to divide by \( x + 2 \), which is equivalent to \( x - (-2) \). Therefore, we will evaluate the polynomial at \( x = -2 \): \[ f(-2) = 3(-2)^{2} - 2(-2) + 4 \] Calculating this step-by-step: 1. \((-2)^{2} = 4\) 2. \(3 \times 4 = 12\) 3. \(-2 \times -2 = 4\) 4. So, \(f(-2) = 12 + 4 + 4 = 20\) Thus, the remainder when \( 3x^{2} - 2x + 4 \) is divided by \( x + 2 \) is \( \boxed{20} \).
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