Let's solve the polynomial division problem: 4x cubed minus 19x squared plus 16x minus 16, divided by x minus 4. We need to divide a cubic polynomial by a linear polynomial. We'll use polynomial long division, which works similarly to numerical long division - we divide term by term systematically. Here's how we set up the division format with the dividend, divisor, and quotient positions clearly labeled.
Now let's perform the first division step. We divide the leading term 4x cubed by x to get 4x squared. This becomes the first term of our quotient. Next, we multiply 4x squared by the entire divisor x minus 4 to get 4x cubed minus 16x squared. We subtract this from the dividend. The 4x cubed terms cancel out, leaving us with negative 19x squared plus 16x squared, which equals negative 3x squared. We bring down the next term 16x to get negative 3x squared plus 16x as our new remainder.
Now for the second division step. We divide negative 3x squared by x to get negative 3x as the next term in our quotient. We multiply negative 3x by the divisor x minus 4 to get negative 3x squared plus 12x. We subtract this from our current remainder negative 3x squared plus 16x. The negative 3x squared terms cancel out, leaving us with 16x minus 12x, which equals 4x. We bring down the final term negative 16 to get 4x minus 16 as our new remainder.
Now for the final division step. We divide 4x by x to get 4 as the final term in our quotient. We multiply 4 by the divisor x minus 4 to get 4x minus 16. We subtract this from our current remainder 4x minus 16. Notice that 4x minus 16 minus 4x minus 16 equals zero. The remainder is zero, which means the division is exact! Our complete quotient is 4x squared minus 3x plus 4.
Let's verify our answer by multiplying x minus 4 times 4x squared minus 3x plus 4. We expand this step by step: x times the quantity 4x squared minus 3x plus 4, minus 4 times the same quantity. This gives us 4x cubed minus 3x squared plus 4x, minus 16x squared plus 12x minus 16. Combining like terms, we get 4x cubed minus 19x squared plus 16x minus 16, which matches our original dividend perfectly. Therefore, our final answer is 4x squared minus 3x plus 4, and since the remainder is zero, x minus 4 is indeed a factor of the original polynomial.