To perform the long division of \( 5x^2 - 13x + 1 \) by \( x - 3 \), follow these steps:
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Divide the first term: Divide the leading term of the numerator (\( 5x^2 \)) by the leading term of the denominator (\( x \)): \[ 5x^2 \div x = 5x. \] This gives the first term of the quotient: \( 5x \).
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Multiply and subtract: Now multiply \( 5x \) by the entire divisor \( x - 3 \): \[ 5x(x - 3) = 5x^2 - 15x. \] Next, subtract this from the original polynomial: \[ (5x^2 - 13x + 1) - (5x^2 - 15x) = -13x + 15x + 1 = 2x + 1. \]
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Repeat the process: Now divide \( 2x \) by \( x \): \[ 2x \div x = 2. \] This gives the second term of the quotient: \( 2 \).
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Multiply and subtract again: Now multiply \( 2 \) by the divisor \( x - 3 \): \[ 2(x - 3) = 2x - 6. \] Subtract this from \( 2x + 1 \): \[ (2x + 1) - (2x - 6) = 1 + 6 = 7. \]
At this stage, there are no more \( x \) terms left to bring down, so we have completed the division.
Thus, the quotient is \( 5x + 2 \) and the remainder is \( 7 \).
Final Answer:
The quotient is \( 5x + 2 \) with remainder \( 7 \).
Check the answer:
To verify, we can reconstruct the original polynomial using the quotient and remainder: \[ (x - 3)(5x + 2) + 7. \] Calculating, \[ (x - 3)(5x + 2) = 5x^2 + 2x - 15x - 6 = 5x^2 - 13x - 6. \] Now adding the remainder: \[ 5x^2 - 13x - 6 + 7 = 5x^2 - 13x + 1. \] This confirms that our long division is correct.
Thus, the answer \( 5x + 2 \) with remainder \( 7 \) is indeed verified.
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