Sing the polynomial function F (x) = x3 - 4x² +x+6 find: a-n
a.standard form: b.factored form: c. leading term: d. x - intercepts & its multiplicities: e. y-intercepts: f. no. of turning points: g. end behavior: h. graph:
a. The standard form of the polynomial function F(x) = x^3 - 4x^2 + x + 6 is already given.
b. To find the factored form, we need to factor the polynomial. However, this particular polynomial does not factor easily. Therefore, the factored form is not readily available.
c. The leading term of the polynomial is x^3, which has the highest degree in the polynomial function.
d. To find the x-intercepts, we set F(x) = 0 and solve for x:
x^3 - 4x^2 + x + 6 = 0
Unfortunately, this polynomial does not have rational roots. Using numerical methods or a graphing calculator, we find that the approximate x-intercepts are:
x ≈ -2.03, x ≈ 0.39, and x ≈ 5.64
The multiplicities of these x-intercepts can be determined by examining the graph of the polynomial.
e. To find the y-intercept, we substitute x = 0 into the polynomial function:
F(0) = (0)^3 - 4(0)^2 + (0) + 6
F(0) = 6
Therefore, the y-intercept is (0, 6).
f. The number of turning points can be determined by examining the graph of the polynomial. In this case, since the degree of the polynomial is odd (3), there will be at most one turning point.
g. The end behavior of the polynomial function F(x) = x^3 - 4x^2 + x + 6 can be determined by looking at the leading term. As x approaches positive infinity, the value of F(x) also approaches positive infinity. As x approaches negative infinity, the value of F(x) also approaches negative infinity.
Answers & Comments
Answer:
a. The standard form of the polynomial function F(x) = x^3 - 4x^2 + x + 6 is already given.
b. To find the factored form, we need to factor the polynomial. However, this particular polynomial does not factor easily. Therefore, the factored form is not readily available.
c. The leading term of the polynomial is x^3, which has the highest degree in the polynomial function.
d. To find the x-intercepts, we set F(x) = 0 and solve for x:
x^3 - 4x^2 + x + 6 = 0
Unfortunately, this polynomial does not have rational roots. Using numerical methods or a graphing calculator, we find that the approximate x-intercepts are:
x ≈ -2.03, x ≈ 0.39, and x ≈ 5.64
The multiplicities of these x-intercepts can be determined by examining the graph of the polynomial.
e. To find the y-intercept, we substitute x = 0 into the polynomial function:
F(0) = (0)^3 - 4(0)^2 + (0) + 6
F(0) = 6
Therefore, the y-intercept is (0, 6).
f. The number of turning points can be determined by examining the graph of the polynomial. In this case, since the degree of the polynomial is odd (3), there will be at most one turning point.
g. The end behavior of the polynomial function F(x) = x^3 - 4x^2 + x + 6 can be determined by looking at the leading term. As x approaches positive infinity, the value of F(x) also approaches positive infinity. As x approaches negative infinity, the value of F(x) also approaches negative infinity.