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The division of \( y^{2} - y - 12 \) by \( y - 4 \) results in \( y + 3 \) with no remainder.
Solución
To divide y² - y - 12 by y - 4, we can first try to factor the quadratic.
Step 1: Factor the quadratic.
We look for two numbers that multiply to -12 and add to -1. These numbers are -4 and +3, so:
y² - y - 12 = (y - 4)(y + 3).
Step 2: Divide by (y - 4).
Since y² - y - 12 = (y - 4)(y + 3), dividing by (y - 4) gives:
[(y - 4)(y + 3)] ÷ (y - 4) = y + 3,
provided that y ≠ 4.
Thus, the quotient is y + 3 and the division has no remainder.
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Simplifique esta solución Bonus Knowledge
To divide \( y^{2} - y - 12 \) by \( y - 4 \), you can use synthetic division or polynomial long division. Let’s use polynomial long division. 1. The leading term of the dividend \( y^2 \) divided by the leading term of the divisor \( y \) gives \( y \). 2. Multiply \( y \) by \( y - 4 \) to get \( y^2 - 4y \). 3. Subtract \( (y^2 - 4y) \) from \( (y^2 - y - 12) \): \((y^2 - y - 12) - (y^2 - 4y) = 3y - 12\). 4. Now divide \( 3y \) by \( y \) to get \( 3 \). 5. Multiply \( 3 \) by \( y - 4 \) to get \( 3y - 12 \). 6. Subtract \( (3y - 12) \) from \( (3y - 12) \) to find the remainder, which is \( 0 \). So, the result is \( y + 3 \) with no remainder.
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