Question

Show that \(x^{4}+3 x^{3}+4 x+6\) is exactly divisible by \(x+1\) Hint: Use factor theorem.

Short answer

To show that the polynomial \(x^4 + 3x^3 + 4x + 6\) is exactly divisible by \(x + 1\), we will use the Factor theorem, which states that if a polynomial f(x) is exactly divisible by (x - c), then f(c) = 0. In this case, we want to test the factor theorem by setting c = -1, as the given factor is in the form of (x - c). Substituting the value of c into the polynomial, we get \((-1)^4 + 3(-1)^3 + 4(-1) + 6 = 1 - 3 - 4 + 6 = -2 + 2 = 0\). Since the result is 0, the Factor theorem verifies that the given polynomial is exactly divisible by \(x + 1\).

Step-by-step solution

State the Factor theorem

Factor theorem states that if a polynomial f(x) is exactly divisible by (x - c), then f(c) = 0.

Set the factor we want to test

We want to test if the polynomial is divisible by \(x + 1\). To do this, we will test the factor theorem by setting c = -1, as our factor is in the form of (x - c).

Plug in the value of c into the polynomial

Substitute the value of c into the polynomial, and check if the result is equal to 0: \((-1)^4 + 3(-1)^3 + 4(-1) + 6 = 1 - 3 - 4 + 6\)

Evaluate the expression

Calculate the expression: \(1 - 3 - 4 + 6 = -2 + 2 = 0\)

Conclusion

Since the result of the polynomial after plugging in -1 is equal to 0, according to the Factor theorem, it verifies that the given polynomial, \(x^4 + 3x^3 + 4x + 6\), is exactly divisible by \(x + 1\).

Key concepts

Polynomial Division

Understanding polynomial division is essential to comprehend how one polynomial can be divided by another, resulting in a quotient and possibly a remainder, much like integer division. In the context of our exercise, when applying the Factor Theorem, we also deal with a form of division. Specifically, we check if a polynomial is divisible by a linear factor with no remainder.

For example, to divide the polynomial \(x^4 + 3x^3 + 4x + 6\) by \(x + 1\), we could use long division, a process that repeatedly subtracts multiples of the divisor from the dividend. It'll involve dividing the leading term of the dividend by the leading term of the divisor and then multiplying the entire divisor by that quotient, subtracting that product from the dividend and repeating the process with the resulting polynomial. This continues until the degree of the remainder is less than the degree of the divisor or until the remainder is zero, which would confirm divisibility and factorization.

Synthetic Division

Synthetic division is a shorthand method of dividing a polynomial by a linear factor of the form \(x - c\) and is particularly useful when dealing with higher-degree polynomials. It simplifies the process by focusing only on the coefficients and the constant term of the divisor.

In our exercise, to perform synthetic division by \(x + 1\), we change the sign of the constant term in the divisor and write \(-1\) to the left of a vertical line. Then, write the coefficients of the polynomial across the top. The process involves bringing down the leading coefficient, multiplying it by \(-1\), and adding it to the next coefficient. This sequence is continued until we reach the constant term. A zero in the last position indicates that the polynomial is divisible by \(x + 1\), aligning with what the Factor Theorem predicts.

Roots of Polynomials

Roots of polynomials are values for which the polynomial equals zero. They are the solutions to the equation \(f(x) = 0\). Roots correspond to the x-intercepts of the polynomial's graph. The Factor Theorem provides a direct way to test potential roots: if substituting a value \(c\) into the polynomial yields zero, then \(c\) is a root, and \(x-c\) is a factor.

Using our exercise as an example, if \(x + 1\) is a factor of the polynomial \(x^4 + 3x^3 + 4x + 6\), then by the Factor Theorem, the root associated with this factor is \(-1\). Our calculation showed that substituting \(-1\) into the polynomial does indeed yield zero, confirming that \(-1\) is a root.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of lower-degree polynomials or factors. If we're able to do that such that the polynomial can be expressed as the product of its factors with no remainder, those factors are considered to be the divisors of the polynomial.

In the context of our exercise, using the Factor Theorem, we've shown that \(x^4 + 3x^3 + 4x + 6\) is exactly divisible by \(x + 1\), implying that it's one of its factors. Finding such factors can eventually lead to the complete factorization of the polynomial, which provides insight on its roots and aids in solving polynomial equations. Complete factorization is most helpful when all the roots of the polynomial are real and rational.