Question

Use the factor theorem and syntl 3x^(4)+4x^(3)-11x^(2)+15x+7;x+3

Answer 1

The question involves using the factor theorem and synthetic division to test if x + 3 is a factor of the polynomial 3x^4 + 4x^3 - 11x^2 + 15x + 7. The factor theorem states that a polynomial f(x) has a factor (x - c) if f(c) = 0. Synthetic division is used to evaluate f(c) by substituting c = -3 into the polynomial.

Step-by-step explanation:

The student is asking how to use the factor theorem and synthetic division to evaluate whether x + 3 is a factor of a given polynomial 3x4 + 4x3 - 11x2 + 15x + 7. According to the factor theorem, if x + 3 is a factor of the polynomial, then the polynomial will equal zero when x = -3. We use synthetic division to test this.

1. Write down the coefficients of the polynomial: 3, 4, -11, 15, 7.
2. Write down the value that we are testing (the root) which in this case is -3.
3. Begin synthetic division by bringing down the first coefficient (3).
4. Multiply this coefficient by the root and write the result under the next coefficient, then add them together and continue the process.
5. If the last number, the remainder, is zero, then x + 3 is indeed a factor of the polynomial.

If after performing synthetic division, the remainder is not zero, then x + 3 is not a factor.