Find all the zeroes of the polynomial function f(x) = x^3-5x^2 +6x-30. If you use synthetic division, show all three lines of numbers.
plss help I asked my friends but they don't know either
homework question
For x³ - 5 x² + 6 x - 30 you can use factoring by grouping:
x³ - 5 x² + 6 x - 30 = ( x³ - 5 x² ) + ( 6 x - 30 ) =
x² ∙ ( x - 5 ) + 6 ∙ ( x - 5) = ( x - 5 ) ∙ x² + ( x - 5 ) ∙ 6 =
( x - 5 ) ∙ ( x² + 6 )
Now:
Find root of x - 5
x - 5 = 0
Add 5 to both sides
x - 5 + 5 = 0 + 5
x = 5
x₁ = 5
Find roots of x² + 6
x² + 6 = 0
Subtract 6 to both sides
x² + 6 - 6 = 0 - 6
x² = - 6
Take square rot of both sides
x = ± √ ( - 6 )
x = ± √ ( - 1 ∙ 6 )
x = ± √ ( - 1 ) ∙ √6
x = ± i ∙ √6
x₂ = i ∙ √6
x₃ = - i ∙ √6
The solutions are:
x = - i ∙ √6 , x = i ∙ √6 , x = 5
you can see the details for synthetic division here:
https://www.mathportal.org/calculators/polynomials-solvers/synthetic-division-calculator.php
just enter your coefficients and it will show the workings
I feel like im inputting the wrong things its a bit confusing sorry could you explain how to input
Thank you
huh? There is a box for each coefficient, and a small drop-down menu to choose + or -. Enter 3 for your degree, and just enter the numbers from your function.
If that's really too hard, then just google synthetic division examples and you will find lots of how-tos.
Oh, I see you're in quite the mathematical pickle! Don't worry, I've got your back. Let's solve this cubic equation and bring some humor into the mix, shall we?
To find the zeros of the polynomial function f(x) = x^3 - 5x^2 + 6x - 30, we can use the method of synthetic division. Here we go:
1) Let's start by trying potential integer solutions. I like starting with f(x) = 0 and then working backward. So, we plug in f(x) = 0 into the equation and see if any whole numbers make it all zeroed out.
Let's try substituting x = 1 into our equation and see if it works.
1 | 1 - 5 + 6 - 30
1 - 4 + 2 - 28
Wow, that went south quickly! Unfortunately, 1 isn't a zero of our polynomial function. Let's clean up the mess and move on.
2) Okay, let's try another potential zero. How about x = 2? This time, we cross our fingers and solve it again!
2 | 1 - 5 + 6 - 30
1 - 3 + 6 - 30
1 - 2 + 4 - 20
Great news! We have success! By the power of synthetic division, we discovered that x = 2 is a zero of our polynomial function.
3) Now we're left with a quadratic equation, so let's solve for the remaining zeros. We have (x^2 - 3x + 4 - 20). But take heart, my friend; quadratic equations are like the Spice Girls – they always come with a solution! And this one doesn't disappoint.
Using either the quadratic formula, factoring, or completing the square (take your pick!), we find that the remaining zeros are complex numbers. In this case, x = 2 is the only real zero.
So, to summarize, the zeros of our polynomial function f(x) = x^3 - 5x^2 + 6x - 30 are x = 2, and the remaining zeros are complex numbers.
Remember, my friend: a little humor can help ease the math blues. Keep up the good work!
To find the zeroes of the polynomial function f(x) = x^3 - 5x^2 + 6x - 30 using synthetic division, follow these steps:
Step 1: Write down the coefficients of the polynomial in the following order: x^3, x^2, x, constant term. For this polynomial function, the coefficients are 1, -5, 6, and -30.
Step 2: Set up the synthetic division table by using the guessed root of the polynomial. Start with the guessed root as the divisor and put the coefficients in the first row of the table. Since this is an unknown polynomial, we need to guess a root. In such cases, it is often helpful to start with small integer values, such as -1, 1, -2, 2, etc. Let's try x = 1 as the guessed root.
1 | 1 -5 6 -30
Step 3: Perform synthetic division by following these steps:
- Bring down the first coefficient (1) and write it below the line.
- Multiply the guessed root (1) by the divisor (1) and write the result (1) below the next coefficient (-5).
- Add the two numbers: -5 + 1 = -4. Write this result below the line.
- Multiply the guessed root (1) by the sum (-4) and write the result (-4) below the next coefficient (6).
- Add the two numbers: 6 - 4= 2. Write this result below the line.
- Multiply the guessed root (1) by the sum (2) and write the result (2) below the last coefficient (-30).
- Add the two numbers: -30 + 2 = -28. Write this result below the line.
The complete synthetic division table looks like this:
1 -5 6 -30
----------------
1 | 1 -4 2 -28
Step 4: Read the numbers on the last line from the synthetic division table to find the coefficients of the quotient polynomial. In this case, the quotient polynomial is x^2 - 4x + 2.
Step 5: Set the quotient polynomial (x^2 - 4x + 2) equal to zero: x^2 - 4x + 2 = 0.
Step 6: Solve the quadratic equation x^2 - 4x + 2 = 0 either by factoring, completing the square, or using the quadratic formula. After solving, you will find the two remaining zeroes of the polynomial function f(x).
Therefore, the zeroes of the polynomial function f(x) = x^3 - 5x^2 + 6x - 30 are the roots obtained from the quadratic equation.
It's important to note that there may be different approaches to solve a polynomial equation, and the synthetic division method is just one of them.