What is the fully factored form (with real coefficients) for the polynomial f(x) = æ® – 1? (Hint: 16 = 1) (x2 – 1)(x4 + æ² + 1) O (x – 1)(x + 1)(x² + x + 1)(x²2 – x + 1). O (2 – 1)* (x + 1)? O (2 – 1) (x + 1)4 O (x – 1) (x + 1)3
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- I am trying to find out if the polynomial 12x^4+8x^3+9x^2+4x+4 is reducible or irreducible over Q. We can't apply Eisenstein's criterion here. But we can try this:12x^4+8x^3+9x^2+4x+4 = x^4+(2/3)x^3+(3/4)x^2+(1/3)x+(1/3) x^4+(2/3)x^3+(3/4)x^2+(1/3)x+(1/3)=(x^2+ax+b)(x^2+cx+d) = x^4+(a+c)x^3+(b+d+ac)x^2+(ad+dc)x+bd We get the equation system: a+c=(2/3)b+d+ac=(3/4)ab+bc=(1/3)bd=(1/3) I get the following result:b=(1/2)d=(2/3)c=(2+sqrt(19))/6 and (2-sqrt(19))/6a=(2/3)-((2+sqrt(19))/6) and (2/3)-((2-sqrt(19))/6) The expression in a contains an irrational number, sqrt(19), this means that the polynomial is irreversible over Q. Is this correct?Find a polynomial of degree n=2 that has the given zero(s) x = −√2, √2If the result of a polynomial division is 2x3 − 4x2 − 5x +23x + 2, make at least threedefinitive statements about the ???????? and the ??????? that is involved. Do you agree with the contention that the functions f(x) = x + 2 and g(x) =x2 − 4x − 2are the same in every respect. Provide evidence to support your position. Do you agree with the statement that “ cubic functions must have at least 1x-intercept but not more than 3, whereas quadratic functions may or may not have x-intercepts.Provide evidence to support your position.
- the polynomial of degree 4, p(x) has leading coefficient 1 had a zero of multiplicity 2 at x=2, a zero of multiplicity 1 at x=0 and a zero of multiplicity 1 at x=-3. find a possible formula for p(x) p(x)=11. Show that f(x) = x5 +2x4-6x3+2x-3 has a solution between 1 and 2. Use the intermediate Value Theorem.What is the coefficient of x^n for the generating function (1)/(1+x)^2?
- The polynomial of degree 5, P(x)P(x) has leading coefficient 1, has roots of multiplicity 2 at x=3x=3 and x=0x=0, and a root of multiplicity 1 at x=−1x=-1Find a possible formula for P(x)P(x).The polynomial of degree 33, P(x)P(x), has a root of multiplicity 22 at x=3x=3 and a root of multiplicity 11 at x=−1x=-1. The yy-intercept is y=−7.2y=-7.2.1. If the result of a polynomial division is 2x3 − 4x2 − 5x + 23 x + 2, make at least three definitive statements about the dividend and the divisor that is involved. 2. Do you agree with the contention that the functions f(x) = x + 2 and g(x) = x2 − 4 x − 2 are the same in every respect. Provide evidence to support your position. 3. Do you agree with the statement that “ cubic functions must have at least 1 x-intercept but not more than 3, whereas quadratic functions may or may not have x-intercepts. Provide evidence to support your position.
- What is the coefficient of the term x^3y^2 expansion of (3x - y2)4?Find a formula for the polynomial P(x)P(x) with degree 3 a root of multiplicity 2 at x=1x=1 a root of multiplicity 1 at x=−4x=-4 yy-intercept (0,−2.8)Is f(x) = x^2 + 1 a standard form or a vertex form or BOTH? OR is it a factored form? (I'm confused because it has 2 terms, not 3 terms.) Is f(x) = 3x^2 - x + 2x - 9 a standard form or none of the above? (I'm confused because it has more than 3 terms.)