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Magazine Chapter 8: Problem 102
For the following problems, divide the polynomials. $$ c^{2}+3 c-88 \text { by } c-8 $$
Short Answer
Expert verified
Question: Divide the given polynomials: \(c^2 + 3c - 88\) divided by \(c - 8\)
Answer: \(c + 5\)
Step by step solution
01
Write the dividend (quadratic) and divisor (linear) polynomials
Write the given polynomials:
Dividend: \(c^2 + 3c - 88\)
Divisor: \(c - 8\)
02
Set up the long division process
Arrange the dividend and divisor as you would in a long division problem. Put the dividend, \(c^2+ 3c - 88\), inside the division bracket and the divisor, \(c - 8\), outside the division bracket.
\[
\begin{array}{c|cc cc}
\multicolumn{2}{r}{c} & +5 \\
\cline{2-5}
c-8 & c^2&+3c&-&88 \\
\cline{2-3}
\multicolumn{2}{r}{c^2} & -8c \\
\cline{2-3}
\multicolumn{2}{r}{} & 11c&-& 88 \\
\end{array}
\]
03
Divide the highest degree term of the dividend by the divisor's highest degree term
Divide the highest degree term of the dividend, \(c^2\), by the highest degree term of the divisor, \(c\). This gives a quotient term of \(c\).
04
Multiply the result by the divisor and subtract it from the dividend
Multiply the quotient term, \(c\), by the divisor, \(c - 8\). This gives the result \(c^2 - 8c\). Write this below the dividend and subtract it from the dividend:
\[
\begin{array}{c|cc cc}
\multicolumn{2}{r}{c} & +5 \\
\cline{2-5}
c-8 & c^2&+3c&-&88 \\
\cline{2-3}
\multicolumn{2}{r}{c^2} & -8c \\
\cline{2-3}
\multicolumn{2}{r}{} & 11c&-& 88 \\
\end{array}
\]
05
Repeat the process
Now, divide the new highest degree term, \(11c\), by the divisor's highest degree term, \(c\). This gives a quotient term of \(+5\).
06
Write the new quotient term and repeat the subtraction process
Multiply the new quotient term, \(+5\), by the divisor, \(c-8\). Write this below the remainder and subtract it from the remainder:
\[
\begin{array}{c|cc cc}
\multicolumn{2}{r}{c} & +5 \\
\cline{2-5}
c-8 & c^2&+3c&-&88 \\
\cline{2-3}
\multicolumn{2}{r}{c^2} & -8c \\
\cline{2-3}
\multicolumn{2}{r}{} & 11c&-& 88 \\
\cline{3-5}
\multicolumn{2}{r}{} & 11c&-& 88 \\
\cline{3-5}
\multicolumn{2}{r}{} & & &0 \\
\end{array}
\]
The remainder after subtracting is zero.
07
Write the final answer
The quotient is \(c + 5\) with a remainder of zero, so the result of the division is:
\[
\frac{c^2 + 3c - 88}{c - 8} = c + 5
\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Long Division
Long division is a fundamental arithmetic operation that is also essential in algebra, especially when dividing polynomials. It helps decompose larger algebraic expressions into simpler forms. To understand polynomial division using long division:
- Start by writing the larger polynomial, known as the dividend, inside the division bracket.
- The smaller polynomial, or the divisor, remains outside the bracket.
- Divide coefficients systematically, one term at a time, beginning with the highest degree term.
- Multiply and subtract sequentially until all terms of the dividend have been divided.
Quadratic Polynomial
A quadratic polynomial is a second-degree polynomial generally in the form \(ax^2 + bx + c\). In simpler terms, a quadratic is an expression with a squared term as the highest exponent. In our division problem, the quadratic polynomial is \(c^2 + 3c - 88\). Here are a few key aspects:
- Quadratics typically produce parabolic graphs which can either open upward or downward.
- Roots of quadratic equations can be found using various methods like factoring, completing the square, or using the quadratic formula.
- Dividing a quadratic polynomial by a linear polynomial often involves simplifying the expression into a first-degree polynomial, along with a remainder.
Linear Divisor
A linear divisor is simply a first-degree polynomial, which usually takes the form \(ax + b\). In our problem, the divisor is \(c - 8\). Here are the features that define a linear divisor:
- Linear polynomials form straight lines when graphed, hence the name "linear."
- The linear divisor essentially reduces the quadratic polynomial to a simpler expression, assuming there is no remainder.
- Division by a linear polynomial is relatively straightforward compared to higher-degree divisors.
Remainder Theorem
The Remainder Theorem provides a useful shortcut in polynomial division. It states that when a polynomial \(f(x)\) is divided by a linear divisor \(x - k\), the remainder of this division is \(f(k)\). For the given exercise, since the remainder is zero after dividing \(c^2 + 3c - 88\) by \(c - 8\), it tells us that \(f(8) = 0\). This indicates:
- The divisor is indeed a factor of the polynomial.
- The polynomial \(f(c)\) evaluates to zero when \(c = 8\).
