Questions: Divide. [ left(6 x^2-46 x+26right) div(x-7) ] Your answer should give the quotient and the remainder. Quotient: Remainder:

Transcript text: Divide. \[ \left(6 x^{2}-46 x+26\right) \div(x-7) \] Your answer should give the quotient and the remainder. Quotient: $\square$ Remainder: $\square$

Solution

Solution Steps

To solve the polynomial division problem, we will use polynomial long division. The process involves dividing the first term of the dividend by the first term of the divisor to get the first term of the quotient. Then, multiply the entire divisor by this term and subtract the result from the dividend. Repeat the process with the new polynomial formed until the degree of the remainder is less than the degree of the divisor. The final result will give us the quotient and the remainder.

Step 1: Set Up the Division

We are tasked with dividing the polynomial $6x^2 - 46x + 26$ by the linear polynomial $x - 7$.

Step 2: Perform Polynomial Long Division

Divide the leading term of the dividend $6x^2$ by the leading term of the divisor $x$ to get the first term of the quotient: \[ 6x^2 \div x = 6x \]
Multiply the entire divisor $x - 7$ by $6x$: \[ 6x(x - 7) = 6x^2 - 42x \]
Subtract this result from the original polynomial: \[ (6x^2 - 46x + 26) - (6x^2 - 42x) = -4x + 26 \]

Step 3: Continue the Division

Now, divide the leading term $-4x$ by $x$: \[ -4x \div x = -4 \]
Multiply the divisor $x - 7$ by $-4$: \[ -4(x - 7) = -4x + 28 \]
Subtract this from the current polynomial: \[ (-4x + 26) - (-4x + 28) = 26 - 28 = -2 \]

Step 4: Compile the Results

The quotient of the division is $6x - 4$ and the remainder is $-2$.

Final Answer

The quotient is $6x - 4$ and the remainder is $-2$. Thus, we can express the final answer as: \[ \boxed{\text{Quotient: } 6x - 4, \text{ Remainder: } -2} \]

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