(a)

the long division method can be used to factor the polynomial 𝑃(𝑥)=𝑥^3−6⋅𝑥^2−7⋅𝑥+60. first, we can write 𝑃(𝑥) divided by (𝑥−5) with a remainder:

𝑃(𝑥)=(𝑥−5)q(𝑥)+r(𝑥)

where q(𝑥) is the quotient and r(𝑥) is the remainder. to find q(𝑥) and r(𝑥), we perform the long division:

markdown

copy code

  x^3 - 6x^2 - 7x + 60

 ____________

    x - 5 

 x^2 - x^2 - x^2 

 ____________

    x^2 - 5x 

 -6x - 7x 

 ____________

     -10x 

  +60 

 ____________

    +60 

thus, 𝑃(𝑥) = (𝑥−5)(𝑥^2−𝑥−12) + 60.

next, we can use the fact that 𝑃(−3)=0 to find the other two roots of 𝑃(𝑥). if 𝑥=−3 is a root of 𝑃(𝑥), then 𝑃(−3)=0. substituting 𝑥=−3 into the factored form of 𝑃(𝑥) gives:

𝑃(−3)=(−3−5)(−3^2−(−3)−12) + 60 = 0

expanding and simplifying the expression inside the parenthesis gives:

0 = −8(−2−9) + 60

0 = −8⋅7 + 60

0 = −56 + 60

0 = 4

since 4 ≠ 0, this means that −3 is not a root of 𝑃(𝑥), which is a contradiction. therefore, 𝑃(−3) ≠ 0, and 𝑥=−3 is not a root of 𝑃(𝑥).

to find the other two roots of 𝑃(𝑥), we can use synthetic division or the rational root theorem. in this case, we will use the rational root theorem, which states that if 𝑎/𝑏 is a root of a polynomial with integer coefficients, then 𝑎 must divide the constant term and 𝑏 must divide the leading coefficient.

since 𝑃(𝑥)=𝑥^3−6⋅𝑥^2−7⋅𝑥+60, the constant term is 60 and the leading coefficient is 1. therefore, the possible rational roots of 𝑃(𝑥) are the divisors of 60, which are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±30, and ±60.

testing these values as roots, we find that 𝑥=6 and 𝑥=−2 are the other two roots of 𝑃(𝑥). therefore

(b)

the factored form of the polynomial function with the specified roots and degrees can be found by using the fact that the degree of the polynomial is equal to the sum of the degrees of the factors.

since the polynomial has simple roots 𝑥=−14, 𝑥=−6, 𝑥=−4, the corresponding factors are (𝑥+14), (𝑥+6), and (𝑥+4), respectively.

since the polynomial has a double root 𝑥=4, the corresponding factor is (𝑥−4)^2.

since the polynomial has a triple root 𝑥=9, the corresponding factor is (𝑥−9)^3.

since the polynomial has a quadruple root 𝑥=14, the corresponding factor is (𝑥−14)^4.

the factored form of the polynomial function is then:

𝑃(𝑥) = −2(𝑥+14)^4(𝑥−14)^4(𝑥+6)(𝑥+4)(𝑥−4)^2(𝑥−9)^3

this is the factored form of the polynomial function with the specified roots and degrees