The solution is \([-4,-1]\cup \left[ {3,\,\infty } \right)\). We see the x-intercept of P(x) is x = 25, as we expected. The roots (zeros) are \(-1+\sqrt{7},\,\,-1-\sqrt{7},\,\,-3\), and \(1\). Remember that if you get down to a quadratic that you can’t factor, you will have to use the Quadratic Formula to get the roots. This tells us a number of things. e)  The dimensions of the open donut box with the largest volume is \(\left( {30-2x} \right)\) by \(\left( {15-2x} \right)\) by (\(x\)), which equals \(\left( {30-2\left( {2.17} \right)} \right)\) by \(\left( {15-2\left( {2.17} \right)} \right)\) by \(\left( {15-2\left( {2.17} \right)} \right)\), which equals 23.66 inches by 8.66 inches by 3.17 inches. Since this function represents your distance from your house, when the function's value is 0, th… And remember that if you sum up all the multiplicities of the polynomial, you will get the degree! Find the other zeros for the following function, given \(5-i\) is a root: Two roots of the polynomial are \(i\) and. Where a function equals zero. When we find the roots of Polynomial Functions, we need to learn how to do synthetic division. For example, if you have the polynomial \(f\left( x \right)=-{{x}^{4}}+5{{x}^{3}}+2{{x}^{2}}-8\), and if you put a number like 3 in for \(x\), the value for \(f(x)\) or \(y\) will be the same as the remainder of dividing \(-{{x}^{4}}+5{{x}^{3}}+2{{x}^{2}}-8\) by \((x-3)\). \right| \,\,\,\,\,2\,\,\,\,\,\,\,\,\,6\,\,\,\,\,\,\,\,k\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,-45\\\underline{{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-6\,\,\,\,\,\,\,\,0\,\,\,\,\,-3k\,\,\,\,\,\,\,\,\,\,\,9k\,\,\,\,\,\,\,\,\,\,\,}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,k\,\,\,\,\,-3k\,\,\,\,\,\left| \! Since volume is \(\text{length }\times \text{ width }\times \text{ height}\), we can just multiply the three terms together to get the volume of the box. Notice how we only see the first two roots on the graph to the left. Let's do both and make sure we get the same result. Use the \(x\) values from the maximums and minimums. To unlock this lesson you must be a Study.com Member. The square root function defined above is evaluated for some nonnegative values of xin the table below. \end{array}. The table below shows how to find the end behavior of a polynomial (which way the \(y\) is “heading” as \(x\) gets very small and \(x\) gets very large). Always try easy numbers, especially 0, if it’s not a boundary point! (a)   Write (as polynomials in standard form) the volume of the original block, and the volume of the hole. In fact, you can even put in, First use synthetic division to verify that, Subtract down, and bring the next digit (, \(x\)  goes into \(\displaystyle {{x}^{3}}\) \(\color{red}{{{{x}^{2}}}}\) times, Multiply the \(\color{red}{{{{x}^{2}}}}\) by “\(x+3\) ” to get \(\color{red}{{{{x}^{3}}+3{{x}^{2}}}}\), and put it under the \({{x}^{3}}+7{{x}^{2}}\). Now this looks really confusing, but it’s not too bad; let’s do some examples. (We’ll learn about this soon). This will give you the value when \(x=0\), which is the \(y\)-intercept). Pretty cool! It makes sense that the root of \({{x}^{3}}-8\) is \(2\); since \(2\) is the cube root of \(8\). This demonstrates a pretty neat connection between algebraic and geometric properties of functions, don't you think? credit by exam that is accepted by over 1,500 colleges and universities. Multiply \(\color{blue}{{4x}}\) by “\(x+3\) ” to get \(\color{blue}{{4{{x}^{2}}+12x}}\), and put it under the \(\displaystyle 4{{x}^{2}}+10x\). 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