Use synthetic division to determine which pair of integers provide both a lower and an upper bound for the zeros of f(x)= x^5-2x^4-25x^3+50x^2+144x-290
PLEASE SHOW WORK!!!
To use synthetic division, we need to find a pair of integers that will provide both a lower and an upper bound for the zeros of the given polynomial function. Here's how you can do it:
Step 1: Identify the leading coefficient and the constant term.
The given polynomial function is f(x) = x^5 - 2x^4 - 25x^3 + 50x^2 + 144x - 290. The leading coefficient is 1 (coefficient of x^5), and the constant term is -290 (coefficient without any x).
Step 2: Find the upper bound.
To find an upper bound, we can look for a positive integer x-value that, when substituted into the polynomial, gives a positive result. Start by substituting x = 1 into the polynomial:
f(1) = (1)^5 - 2(1)^4 - 25(1)^3 + 50(1)^2 + 144(1) - 290
= 1 - 2 - 25 + 50 + 144 - 290
= -122
Since the result is negative (-122), 1 is not an upper bound.
Now, try substituting x = 2:
f(2) = (2)^5 - 2(2)^4 - 25(2)^3 + 50(2)^2 + 144(2) - 290
= 32 - 32 - 200 + 200 + 288 - 290
= -2
Since the result is also negative (-2), 2 is not the upper bound either.
Keep trying larger positive integers until you find a positive result. Let's try x = 3:
f(3) = (3)^5 - 2(3)^4 - 25(3)^3 + 50(3)^2 + 144(3) - 290
= 243 - 162 - 2025 + 450 + 432 - 290
= -340
Again, the result is negative (-340). So, 3 is not the upper bound.
Continue this process until you find the smallest positive integer where f(x) produces a positive result. Let's try x = 4:
f(4) = (4)^5 - 2(4)^4 - 25(4)^3 + 50(4)^2 + 144(4) - 290
= 1024 - 512 - 6400 + 800 + 576 - 290
= -4298
The result is negative (-4298). Therefore, 4 is not the upper bound.
By continuing this process and trying larger positive integers, we find that f(5) is positive:
f(5) = (5)^5 - 2(5)^4 - 25(5)^3 + 50(5)^2 + 144(5) - 290
= 3125 - 1250 - 3125 + 1250 + 720 - 290
= 1230
Since f(5) gives a positive result (1230), we can conclude that 5 is the upper bound.
Step 3: Find the lower bound.
To find a lower bound, we need to find a negative integer x-value that, when substituted into the polynomial, gives a positive result. Follow the same process as in Step 2, but this time, try negative integer values.
Start with x = -1:
f(-1) = (-1)^5 - 2(-1)^4 - 25(-1)^3 + 50(-1)^2 + 144(-1) - 290
= -1 - 2 + 25 + 50 - 144 - 290
= -362
The result is negative (-362). So, -1 is not a lower bound.
Now try x = -2:
f(-2) = (-2)^5 - 2(-2)^4 - 25(-2)^3 + 50(-2)^2 + 144(-2) - 290
= -32 - 32 + 200 + 200 - 288 - 290
= -342
Again, the result is negative (-342). Hence, -2 is not the lower bound.
Continue trying larger negative integers until you find a positive result. Let's try x = -3:
f(-3) = (-3)^5 - 2(-3)^4 - 25(-3)^3 + 50(-3)^2 + 144(-3) - 290
= -243 - 162 + 2025 + 450 - 432 - 290
= 348
The result is positive (348). Therefore, -3 is the lower bound.
Hence, the pair of integers that provide both a lower and an upper bound for the zeros of the given function f(x) = x^5 - 2x^4 - 25x^3 + 50x^2 + 144x - 290 is (-3, 5).