Join IDNLearn.com to access a wealth of knowledge and get your questions answered by experts. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.
Sagot :
We have to factorize the polynomial:
[tex]2x^3+10x^2-28x[/tex]using the gratest common factor (GFC).
Then, we have to find all x-intercepts of the polynomial.
We can factorize each term as:
[tex]\begin{gathered} 2x^3=2\cdot x^3 \\ 10x^2=2\cdot5\cdot x^2 \\ 28x=2\cdot2\cdot7\cdot x=2^2\cdot7\cdot x \end{gathered}[/tex]We have 2 and x as common factors of the polynomial, so we write:
[tex]\begin{gathered} 2x^3+10x^2-28x \\ 2x(x^2)+2x(5x)-2x(2\cdot7) \\ 2x(x^2+5x-14) \end{gathered}[/tex]We now need to apply the quadratic formula to find the roots of the quadratic polynomial in parenthesis:
[tex]\begin{gathered} x^2+5x-14 \\ \Rightarrow x=\frac{-5\pm\sqrt[]{5^2-4\cdot1\cdot(-14)}}{2\cdot1} \\ x=\frac{-5\pm\sqrt[]{25+56}}{2} \\ x=\frac{-5\pm\sqrt[]{81}}{2} \\ x=\frac{-5\pm9}{2} \\ x_1=\frac{-5-9}{2}=\frac{-14}{2}=-7 \\ x_2=\frac{-5+9}{2}=\frac{4}{2}=2 \end{gathered}[/tex]We can now factorize the polynomial as:
[tex]x(x^2+5x-14)=x(x+7)(x-2)[/tex]This factorized form gives us the x-intercepts:
[tex]\begin{gathered} f(x)=0 \\ x(x+7)(x-2)=0 \\ \Rightarrow x_1=0 \\ x_2=-7 \\ x_3=2 \end{gathered}[/tex]Answer:
The factorized polynomial is x(x+7)(x-2).
The x-intercepts are x = 0, x = -7 and x =2.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and see you next time for more reliable information.
