Video Transcript
Find the quotient when negative six π₯ cubed plus 11π₯ squared plus π₯ minus six is divisible by π₯ minus one.
We can find an expression for negative six π₯ cubed plus 11π₯ squared plus π₯ minus six divided by π₯ minus one using long division. The first step is to find the quotient of the leading terms of the dividend and the divisor. We note that negative six π₯ cubed divided by π₯ is equal to negative six π₯ squared. So, we write negative six π₯ squared in the quotient.
Next, we need to subtract negative six π₯ squared times the divisor from the dividend. So next, we find that negative six π₯ squared times the divisor, π₯ minus one, equals negative six π₯ cubed plus six π₯ squared. This will be subtracted from the dividend, so we line up the like terms in columns and then subtract this expression.
We must be very careful to subtract each term, not just the first term. This is why we use parentheses. It may also be helpful to think of this step as distributing a negative through the polynomial and then combining like terms. This looks like adding six π₯ cubed and subtracting six π₯ squared from the dividend. Then, the cubic terms cancel. And we are left with five π₯ squared plus π₯ minus six after combining the like terms.
We call this expression the remainder. However, as long as the degree of the remainder is greater than or equal to the degree of the divisor, then we can still perform another round of division. We recall that when two polynomials are divisible, their remainder is zero. So, in this case, we expect a zero remainder at the end of our long division.
Letβs pause to clarify what we have determined so far. We have shown that negative six π₯ cubed plus 11π₯ squared plus π₯ minus six divided by π₯ minus one is equal to negative six π₯ squared plus five π₯ squared plus π₯ minus six over π₯ minus one, where five π₯ squared plus π₯ minus six is our remainder and π₯ minus one is our divisor. Since the degree of the remainder is still greater than the degree of the divisor, we have more rounds of division to perform. This means that the remainder five π₯ squared plus π₯ minus six is the new dividend.
In this next round, the quotient of the leading term of the dividend and the leading term of the divisor is five π₯. So, we add five π₯ to the quotient. Then, we must multiply the divisor by five π₯, which equals five π₯ squared minus five π₯. Then, we subtract this expression from the new dividend. We line up the like terms in columns and must be very careful to subtract each term, not just the first term. It may be helpful to think of this step as distributing a negative through the polynomial and then combining like terms. This looks like subtracting five π₯ squared and adding five π₯ to the dividend. Then, we are left with a remainder of six π₯ minus six.
In summary, so far, we have shown that negative six π₯ cubed plus 11π₯ squared plus π₯ minus six divided by π₯ minus one is equal to negative six π₯ squared plus five π₯ plus six π₯ minus six over π₯ minus one. Since the degree of the newest dividend and the divisor are equal, we can still perform one more round of division. This begins with finding the quotient of the leading terms of the dividend and the divisor, which is equal to six. So, we add six to the quotient. Then, we find the product of six and the divisor to subtract from the new dividend.
Lining up the like terms in columns, we subtract six π₯ minus six from the dividend, being careful to subtract each term. We finally reach the zero remainder. Therefore, the quotient when negative six π₯ cubed plus 11π₯ squared plus π₯ minus six is divisible by π₯ minus one is negative six π₯ squared plus five π₯ plus six.
An alternative way of completing this division would have been to factor the highest common factor, six, from the new dividend. Then, dividing this factored expression by π₯ minus one simplifies to give us the last term of the quotient, six. So, regardless of which way we finish the division, we still find that the quotient is negative six π₯ squared plus five π₯ plus six.
