91Ó°ÊÓ

What is the corresponding binomial factor of a polynomial, \(P(x),\) given the value of the zero? a) \(P(1)=0\) b) \(P(-3)=0\) c) \(P(4)=0\) d) \(P(a)=0\)

What are the degree, type, leading coefficient, and constant term of each polynomial function? a) \(f(x)=-x+3\) b) \(y=9 x^{2}\) \(g(x)=3 x^{4}+3 x^{2}-2 x+1\) d) \(k(x)=4-3 x^{3}\) e) \(y=-2 x^{5}-2 x^{3}+9\) f) \(h(x)=-6\)

Determine each quotient, \(Q\), using synthetic division. a) \(\left(x^{3}+x^{2}+3\right) \div(x+4)\) b) \(\frac{m^{4}-2 m^{3}+m^{2}+12 m-6}{m-2}\) c) \(\left(2-x+x^{2}-x^{3}-x^{4}\right) \div(x+2)\) d) \(\left(2 s^{3}+3 s^{2}-9 s-10\right) \div(s-2)\) e) \(\frac{h^{3}+2 h^{2}-3 h+9}{h+3}\) f) \(\left(2 x^{3}+7 x^{2}-x+1\right) \div(x+2)\)

What are the possible integral zeros of each polynomial? a) \(P(x)=x^{3}+3 x^{2}-6 x-8\) b) \(P(s)=s^{3}+4 s^{2}-15 s-18\) c) \(P(n)=n^{3}-3 n^{2}-10 n+24\) d) \(P(p)=p^{4}-2 p^{3}-8 p^{2}+3 p-4\) \(P(z)=z^{4}+5 z^{3}+2 z^{2}+7 z-15\) f) \(P(y)=y^{4}-5 y^{3}-7 y^{2}+21 y+4\)

Use the remainder theorem to determine the remainder when each polynomial is divided by \(x+2\) a) \(x^{3}+3 x^{2}-5 x+2\) b) \(2 x^{4}-2 x^{3}+5 x\) c) \(x^{4}+x^{3}-5 x^{2}+2 x-7\) d) \(8 x^{3}+4 x^{2}-19\) e) \(3 x^{3}-12 x-2\) f) \(2 x^{3}+3 x^{2}-5 x+2\)

A snowboard manufacturer determines that its profit, \(P,\) in dollars, can be modelled by the function \(P(x)=1000 x+x^{4}-3000\) where \(x\) represents the number, in hundreds, of snowboards sold. a) What is the degree of the function \(P(x) ?\) b) What are the leading coefficient and constant of this function? What does the constant represent? c) Describe the end behaviour of the graph of this function. d) What are the restrictions on the domain of this function? Explain why you selected those restrictions. e) What do the \(x\) -intercepts of the graph represent for this situation? f) What is the profit from the sale of 1500 snowboards?

For each function, determine i) the \(x\) -intercepts of the graph ii) the degree and end behaviour of the graph iii) the zeros and their multiplicity iv) the \(y\) -intercept of the graph v) the intervals where the function is positive and the intervals where it is negative a) \(y=x^{3}-4 x^{2}-45 x\) b) \(f(x)=x^{4}-81 x^{2}\) c) \(h(x)=x^{3}+3 x^{2}-x-3\) d) \(k(x)=-x^{4}-2 x^{3}+7 x^{2}+8 x-12\)

Determine the value(s) of \(k\) so that the binomial is a factor of the polynomial. a) \(x^{2}-x+k, x-2\) b) \(x^{2}-6 x-7, x+k\) c) \(x^{3}+4 x^{2}+x+k, x+2\) d) \(x^{2}+k x-16, x-2\)

Without using technology, sketch the graph of each function. Label all intercepts. a) \(f(x)=x^{4}-4 x^{3}+x^{2}+6 x\) b) \(y=x^{3}+3 x^{2}-6 x-8\) c) \(y=x^{3}-4 x^{2}+x+6\) d) \(h(x)=-x^{3}+5 x^{2}-7 x+3\) e) \(g(x)=(x-1)(x+2)^{2}(x+3)^{2}\) f) \(f(x)=-x^{4}-2 x^{3}+3 x^{2}+4 x-4\)

Populations in rural communities have declined in Western Canada, while populations in larger urban centers have increased. This is partly due to expanding agricultural operations and fewer traditional family farms. demographer uses a polynomial function to predict the population, \(P,\) of a town \(t\) years from now. The function is \(P(t)=t^{4}-20 t^{3}-20 t^{2}+1500 t+15000\) Assume this model can be used for the next 20 years.a) What are the key features of the graph of this function? b) What is the current population of this town? c) What will the population of the town be 10 years from now? d) When will the population of the town be approximately 24 000?