Question

1.) Which statements are true about the polynomial function? f(x)=3x^3-4x^2-13x-6

• (x-3) is a factor of f(x)
• f(x)=0 when x=–3
• (3x+2) is a factor of f(x)
• f(-1)=0
• f(x) divided by (x+1) has a remainder of 0
• f(x)=0 when x=3
• (3x-2) is a factor of f(x)

2.) What are the factors of the polynomial function? f(x)=x^3-x^2-10x-8 Possible rational roots ±p/q: ±1, ±2, ±4, ±8.
• (x+1)
• (x-1)
• (x+2)
• (x-2)
• (x+4)
• (x-4)
• (x-5)

3.) One of the zeros of the polynomial function is 5. f(x)=x^4-4x^3-6x^2+4x+5. What is the factored form of the function?
• (x-5)(x+5)(x-1)(x+1)
• (x-5)(x-1)(x+1)^2
• (x-5)(x+1)(x-1)^2
• (x-1)(x+1)(x-5)^2

Answer 1

3.) •
`\displaystyle (x - 5)(x - 1)(x + 1)^2`

2.)
`\displaystyle (x - 4), (x + 2), and\:(x + 1)`

1.) • f(x) = 0 when x = 3

• f(x) divided by (x + 1) has a remainder of 0

• f(−1) = 0

• (3x + 2) is a factor of f(x)

• (x - 3) is a factor of f(x)

Step-by-step explanation:

3.) By the Rational Root Theorem, we would take the Least Common Divisor [LCD] between the leading coefficient of 1, and the initial value of 5, which is 1, but we will take 5 it is a rational number we can work with; so this automatically makes our first factor of
`\displaystyle x - 5`. Next, since the factor\divisor is in the form of
`\displaystyle x - c`, use what is called Synthetic Division. Remember, in this formula, −c gives you the OPPOSITE terms of what they really are, so do not forget it. Anyway, here is how it is done:

5| 1 −4 −6 4 5

↓ 5 5 −5 −5

_____________

1 1 −1 −1 0 →
`\displaystyle x^3 + x^2 - x - 1`

You start by placing the c in the top left corner, then list all the coefficients of your dividend [x⁴ - 4x³ - 6x² + 4x + 5]. You bring down the original term closest to c then begin your multiplication. Now depending on what symbol your result is tells you whether the next step is to subtract or add, then you continue this process starting with multiplication all the way up until you reach the end. Now, when the last term is 0, that means you have no remainder. Finally, your quotient is one degree less than your dividend, so that 1 in your quotient can be an x³, the x² follows right behind it, bringing −x right up against it, and bringing up the rear, −1, giving you the quotient of
`\displaystyle x^3 + x^2 - x - 1.`

However, we are not finished yet. This is our first quotient. The next step, while still using the Rational Root Theorem with our first quotient, is to take the Greatest Common Divisor [GCD] of the leading coefficient of 2, and the initial value of −1, which is 1, so this makes our next factor of
`\displaystyle x - 1`. Then again, we use Synthetic Division because
`\displaystyle x + 1`is in the form of
`\displaystyle x - c`:

1| 1 1 −1 −1

↓ 1 2 1

_________

1 2 1 0 →
`\displaystyle x^2 + 2x + 1 >> (x + 1)^2`

So altogether, we have our four factors of
`\displaystyle (x - 5)(x + 1)^2(x - 1).`

_______________________________________________

2.) By the Rational Root Theorem again, this time, we will take 4,since the initial value is −8. This gives our automatic factor of
`\displaystyle x - 4.`Then start up Synthetic Division again:

4| 1 −1 −10 −8

↓ 4 12 8

___________

1 3 2 0 →
`\displaystyle x^2 + 3x + 2 >> (x + 1)(x + 2)`

So altogether, we have our three factors of
`\displaystyle (x + 1)(x + 2)(x - 4).`

_______________________________________________

1.) By the Rational Root Theorem one more time, this time, we will take 3 since the initial value is −6. This gives our automatic factor of
`\displaystyle x - 3.`Then start up Synthetic Division again:

3| 3 −4 −13 −6

↓ 9 15 6

____________

3 5 2 0 →
`\displaystyle 3x^2 + 5x + 2`

Finally, you can just simply factor this second quotient:

`\displaystyle 3x^2 + 5x + 2 \\ \\ (3x^2 + 2x) + (3x + 2) \\ x(3x + 2) \: \: 1(3x + 2) \\ \\ (x + 1)(3x + 2)`

So altogether, we have our three factors of
`(x + 1)(x - 3)(3x + 2),`and when set to equal zero, you will get
`\displaystyle -(2)/(3), 3, and\:1.`With all the information given, you should be capable of figuring out the true statements.

I am joyous to assist you anytime.

Answer 2

1.

• (x -3) is a factor of f(x)
• (3x+2) is a factor of f(x)
• f(-1) = 0
• f(x) divided by (x+1) has a remainder of 0
• f(x) = 0 when x = 3

2.

• (x+1)
• (x+2)
• (x-4)

3.

• (x-5)(x-1)(x+1)^2

Explanation:

Each of these polynomials has a sum of odd-degree coefficients equal to the sum of even-degree coefficients. This means -1 is a root. The cubics can be reduced to quadratics by removing those factors of (x+1).

In any event, I like to use a graphing calculator to show me the roots of higher degree polynomials.

_____

1. Factoring out (x+1) reduces the cubic to 3x^2 -7x -6, which will have a factor of (x -3). Dividing that out gives the remaining factor of 3x+2.

So, the factorization is f(x) = (x+1)(3x+2)(x-3). It will have zeros at -1, -2/3 and 3. Knowing these things can help you choose the correct true statements from the list.

__

2. Factoring out (x+1) reduces the cubic to x^2 -2x -8. Factoring that tells you the factorization is f(x) = (x+2)(x+1)(x-4). This is all you need to know to pick the correct factors from the list.

__

3. Factoring out (x-5) reduces the quartic to x^3 +x^2 -x -1. The sum of coefficients is zero, so you know x=1 is a root. Also, the sums of odd-degree and even-degree coefficients are the same (0), so you know that -1 is also a root. Factoring out known roots gives you a remaining factor of (x+1), so the factorization is f(x) = (x-5)(x-1)(x+1)^2.

_____

Comment on factoring the quadratic in Problem 1

Maybe you have learned to factor quadratics like 3x^2 -7x -6. The job involves finding factors of (3)(-6) = -18 that have a sum of -7. Those factors would be -9 and 2. With this knowledge, you can write the factorization as ...

(3x-9)(3x+2)/3 . . . . . all of the 3s in this expression are copies of the leading coefficient.

The factor (3x-9) is the only one that conveniently divides by 3, so this expression reduces to (x-3)(3x+2).

Of course, you can find the roots of the quadratic however you like if factorization doesn't work well for you.