Problem 102 - Elementary Algebra

Question

For the following problems, divide the polynomials. $$ c^{2}+3 c-88 \text { by } c-8 $$

Step-by-Step Solution

Answer

Question: Divide the given polynomials: \(c^2 + 3c - 88\) divided by \(c - 8\) Answer: \(c + 5\)

Step 1: Write the dividend (quadratic) and divisor (linear) polynomials

Write the given polynomials: Dividend: \(c^2 + 3c - 88\) Divisor: \(c - 8\)

Step 2: 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. \[ \begin{array}{c|cc cc} \multicolumn{2}{r}{c} & +5 \\ \cline{2-5} c-8 & c^2&+3c&-&88 \\ \cline{2-3} \multicolumn{2}{r}{c^2} & -8c \\ \cline{2-3} \multicolumn{2}{r}{\underline{\phantom{xx}}} & 11c&-& 88 \\ \end{array} \]

Step 3: 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 \(c\).

Step 4: 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: \[ \begin{array}{c|cc cc} \multicolumn{2}{r}{c} & +5 \\ \cline{2-5} c-8 & c^2&+3c&-&88 \\ \cline{2-3} \multicolumn{2}{r}{c^2} & -8c \\ \cline{2-3} \multicolumn{2}{r}{\underline{\phantom{xx}}} & 11c&-& 88 \\ \end{array} \]

Step 5: 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 \(+5\).

Step 6: 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: \[ \begin{array}{c|cc cc} \multicolumn{2}{r}{c} & +5 \\ \cline{2-5} c-8 & c^2&+3c&-&88 \\ \cline{2-3} \multicolumn{2}{r}{c^2} & -8c \\ \cline{2-3} \multicolumn{2}{r}{\underline{\phantom{xx}}} & 11c&-& 88 \\ \cline{3-5} \multicolumn{2}{r}{\underline{\phantom{xx}}} & 11c&-& 88 \\ \cline{3-5} \multicolumn{2}{r}{\underline{\phantom{xx}}} & & &0 \\ \end{array} \] The remainder after subtracting is zero.

Step 7: Write the final answer

The quotient is \(c + 5\) with a remainder of zero, so the result of the division is: \[ \frac{c^2 + 3c - 88}{c - 8} = c + 5 \]

Key Concepts

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.

This process mirrors the long division method used in basic arithmetic, allowing for a step-by-step breakdown of components. Break it into manageable segments to simplify complex polynomial expressions.

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.

Understanding these features allows us to handle quadratic expressions more effectively in various mathematical contexts.

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.

Utilizing a linear divisor in polynomial division, like in our exercise, simplifies solving by reducing the complexity of polynomials.

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 \(c = 8\).

Using the Remainder Theorem can dramatically reduce the work required to check factors or confirm divisions, making it a powerful tool in algebraic calculations.