So for example, when X is equal to three, H of X is equal to zero. So you could use this asĪ definition of G of T. We map between different values of T and what G of T would be. Voiceover:So we have three different function definitions here. The quadratic formula yields roots 3 ± √5.
We can continue to search for roots by finding the roots of the quadratic: From our analysis above, we know that (x - 1) is a factor of the polynomial, so we want to divide the polynomial by (x - 1) and find the quotient. Here is the systematic algebraic way to do it:
Your second question asks if there is an easier way to solve the following equation: Therefore (x - 1) is indeed a factor of 2x³ - 14x² + 20x - 8. Which we note is 0, because the first 3 terms are from the original function ƒ(x) and that already yielded 8, and when we combine that with the remaining -8, we get 0. If we divide by (x - 1) our remainder is: Since the remainder is 8 and we want to get rid of that, we subtract 8 to get: We can make (x - 1) a factor of ƒ(x) if we add something to the function that will get rid of the remainder. It follows that if we divide ƒ(x) by (x - 1), then our remainder is 8. Remainder Theorem tells us that when we divide ƒ(x) by a linear binomial of the form (x - a) then the remainder is ƒ(a).
That means when you plug in 1 for "x" in the above expression, you will get 8.
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