Question

Find the constant c such that the denominator will divide evenly into the numerator. $$\frac{x^{5}-2 x^{2}+x+c}{x+2}$$

Short answer

The constant c that will allow the denominator to divide evenly into the numerator is 42.

Step-by-step solution

Substitute \(x = -2\) into the equation

By the factor theorem, if \((x+2)\) is a factor of \(x^{5}-2 x^{2}+x+c\), then substituting \(x=-2\) into the equation should simplify it to 0. So, \((-2)^5 - 2(-2)^2+(-2)+c = 0.\)

Simplify the equation

This simplifies to \(-32 - 8 - 2 + c = 0\). If you continue to simplify this, it simplifies to \(c = 42\).

Verify \((x+2)\) is a factor

We can confirm that \((x+2)\) is a factor of \((x^5 - 2x^2 + x + 42)\) by performing polynomial division or by using the synthetic division method. Either method will confirm that the division is exact, hence demonstrating that \((x+2)\) is indeed a factor and our solution for \(c\) is correct.

Key concepts

Factor Theorem

The Factor Theorem is a powerful tool we can use to determine if a specific binomial, like \((x - a)\), is a factor of a polynomial.
It's based on a simple yet profound idea: if \((x - a)\) is a factor of the polynomial \(f(x)\), then substituting \(a\) into \(f(x)\) should give you zero.
Imagine you have a polynomial \(f(x) = x^{5}-2x^{2}+x+c\) and we want to check if \(x+2\) is a factor.
According to the Factor Theorem, substitute -2 into the equation:
\((-2)^{5} - 2(-2)^{2} + (-2) + c = 0\). After simplifying, if the result is zero, \((x+2)\) is indeed a factor.

• The Factor Theorem connects the idea of roots and factors.
• This method is essential for confirming whether a number makes the entire polynomial zero.

Thus, it helps in solving polynomial factorization problems efficiently.

Synthetic Division

Synthetic Division is a simplified form of polynomial division. It's particularly used to divide polynomials by linear factors such as \(x-a\).
This method is quick and reduces the amount of work involved compared to traditional division.
Consider our example, where we want to divide \(x^5 - 2x^2 + x + 42\) by \(x+2\). While long division is possible, synthetic division is often faster and less tedious.To start synthetic division:

• Identify the zero of the divisor \(x+2\), which is \(-2\).
• Write the coefficients of the polynomial: [1, 0, 0, -2, 1, 42].
• Bring down the leading coefficient and multiply it by \(-2\); place the result in the next column and add.

Repeat these steps across the coefficients to yield the quotient and remainder.
If the remainder is zero, \(x+2\) is a factor.

Polynomial Factorization

Polynomial Factorization involves breaking down a complex polynomial into simpler polynomials that, when multiplied, reproduce the original polynomial.
This process is crucial in solving equations, graphing functions, and understanding polynomial behavior.For the polynomial \(x^5 - 2x^2 + x + 42\), factorization requires finding values for which the polynomial evaluates to zero. Using the synthetic division result, if \((x + 2)\) divides evenly into it, this factor can be further explored to see if more factors can be pulled out.

• Start by identifying possible roots using the Factor Theorem and synthetic division.
• Each divisor that results in zero indicates a factor of the polynomial.
• The original polynomial equals the product of these factors.

Understanding factorization can help in finding polynomial roots and solving equations by simplifying them into manageable parts.